THEORY OF CAPACITIES (')
by Guatavo CI'IOQLIIT(:)(a).
INTRODUCTION
This work originated frown 'the following problem, whose
significance had been emphasized by M. Brelot and H. Cartan:
Is the interior Newtonian capacity of an arbitrary borelian
subset X of the space R equal to the exterior Newtonian
capacity of X ?
For the solution of this problem, I first systematically
studied the non-additive set-functions, and tried to extract
from their totality certain particularly interesting classes,
with a view to establishing for t.h.ese a theory analogous to
the classical theory of measurabhty.
I succeeded later in showing that the classical Newtonian
capacity œ belongs to one of these classes, more precisely: if A
and B are arbitrary compact subsets. Of R , then
f(A U B) + f(A 13 B)f(A) +
It lolløwed from 'this that every borelian, and ev.en every
analytic set ' capacitable with respect to the Newroman capa-
s
city, a result which can, moreover, be extended to the capa-
(J) This research was supported by the United States Air Force, throught the
Office of Scientific Research of the Air Research and Development Command.
Ce travail a 6t6 ntis au point en anglais h la suite de conf6rences en anglais; une
version fran9aise n'aurait pu tre pr.te h temps pour l'insertion dans ce volume.
Ie petit article qui suit, comp16tant celui-ci, est naturellement publi6 dans la
mme langue. (Note de la R6daction).
(a) Visiting Research Professor of Mathematics, University of Kansas, Lawrence,
Kansas, 1953-1954.
(3) I wish to express ny thanks to Professor G. B. Price for the very valuable
help which he extended to me in connection with establishing the final version of
the English text; thanks are due also to G. Ladnet, K. Lucas, and E. McLachlan
for their untiring collaboration.
i32 avs?v COQVr
cities associated with the Green's function and to other clas-
sical capacities.
The above inequality, which may be called the inequality of
strong sub-additivity, is equivalent to the following:
V,(X; A, B) -:. f(x) ... /(x O A) ../(X U B)q-/(x O A U B)0.
Now, this relation is the first of an infinite 'sequence of inde-
pendent inequalities, each of the form ¾,(X; A,, A., ..., A)__<0,
which expresses the fact that the successive differences in
an obvious sense of the function œ are alternately positive
or negative.
Thus, the NewtonJan capacity is seen to be an analogue of
the functions of a real variable whose successive derivatives
are alternately positive and negative.
It is known from a theorem of S. Bernstein that these
functions,. termed completely monotone, have an integral
representation in terms of functions e -. Likewise, the set
functions which are ½½ alternating of order infinity possess
an integral representation in terms of exponentials, that is,
of set functions (X) which satisfy
0 +..< t and +(X U Y)
These exponentials take values in [0, 1] only, and this makes it
possible to give a remarkable probabilistic interpretation of
the functions which are alternating of order infinity.
More generally, a detailed study of several other classes of
functions justifies the interest in the determination of 'the
extremal elements of convex cones of functions, and in the
utilization of the corresponding integral representations.
CHAPTER I. Borelian and analytic sets in topololical
spaces. In this chapter, bore]ian and analytic subsets of
arbitrary Hausdorff spaces are redefined and studied. In
fact, a mere adaptation of the classical definitions would
lead to .sets of an irregular topological character for which
an interesting theory Of capacitability could not be construc-
ted. Therefore, we designate as borelian and analytic sets
the sets which are generated by beginning with the compact
sets and using the operations of countable intersection and
union, and continuous mapping (or projection) only. Thus
T.O¾ o cCT.s i33
the operations of difference or complementation are
not used.
The role which the G, sets play in the classical theory is here
played by the K.:; sets.
CHAPTER II. Newtonian and greenian capacities. In
this chtpter, the' Newtonian and Greenian capacities of
c.o.mpact sets. and, thereafter, the interior and exterior capa-
cities of arbitrary sets, are defined. An equilibrium poten-
tial h(X), and a capacity, /(X), are associated with each
compact subset X of a domain. The successive differences
( i)V(x; A,,..., A,) are defined for each of these func-
tions; it is shown that each Va,is non-positive, and the Condi-
tions fo r the vanishing of the V are determined.
It is shown that the sequence of these inequalities for the
capacity [ is complete in the sense that every inequality
between the capacities of a family of compact sets obtained
from p arbitrary compact sets by the operation of union is a
consequence of the inequalities V,..<0. A more penetra-
ting analysis shows that this result remains valid for the capa-
cities of the sets which are obtained from p arbitrary compact
sets by means of the operations of union, intersection, and
difference.
From the relation <0, the following inportant inequality
is obtained:
f(UA,) f(Ua,)E A,)-f(a,)],
for every finite or countable family couples of compact sets
a and A.satisfying the relation ac A for each i.
Furthermore, from the relation V..<. 0, we deduce a result
concerning the capacity of certain compact sets, relative to
domains which are invariant under a one-parameter group of
euclidian motions.
The chapter ends with the study of the differential of/(K)
with respect to suitable increments AK .of K, and with the
derivation of a formula which shows that the Green's function
G(P,, P.) of a domain is a limit of the function
G(K, K) . .f.(K,) +f(Kd f(K, U K.)
' 2f(K,).f(Kd '
134 ousrxv
CHAPTER III. &Itcmating and monotone functions. Capa-
cities. This chapter introduces several classes of fun,u-
t{ons and certain basic concepts as follows: alternating
(monotone) functions of order n or which are mappings
from a commutatlve semi-group into an ordered commu-
tative group and which satisfy
k-7 < 0 \-7- 0-' the conce t of
\ P. _ P ..... ;; P
connected by the relation
inequalities
conjugate
of the form
set functions,
9'(X') -{- (X) .--0, where X" I X is the complement' of X
the concepts of capacity on a class of subsets of a topological
space, of interior capacity ([,) and exterior capacity (f.*), of
alternating and monotone capacities, of a capacity which is
the conjugate of another capacity.
CHAPTER IV. Extension and restriction of a capacity.
The extension f. of a capacity /,, defined on a class g, of
subsets of a space E, to a class g properly containing g,,
by means of the equality /,.(X)-/(X), can often be .used
as a means for. regularizing the class g, and also as a means of
simplifying proofs of capacitability.
On the other hand, the operation of restriction will some-
times make it possible to replace the space E by a simpler
space.
Furthermore, the preservation of various classes of capa-
cities (alternating or monotone) under these operations of
extension and restriction is studied; for instance, let g,.be the
class of all compact subsets of a Hausdorff space E; then, if
[, is alternating of order n, the same is true for every exten-
sion f, of h.
CHAPTER V. Operations on capacities and examples of
capacities. , First, several operations which leave certain
classes of capacities invariant are studied: 'for instance, if
a capacity g(Y), alternating of order n, is defined on the class
3;(F) of all compact subsets of a spCnetoF , and if Y- ?(X)
denotes a mapping from 3(E) 3(F) such that
(X, u X) (X,) u ?(X) and which satisfies, in addition,
a certain requirement of continuity on the right, then the
THEORY OF CAPACITIES
135
function œ(X)- g(Y) is also a capacity which is alterna-
ting of order n. The projection operation is such an operation
and will play an essential role in the study of capacitability.
The remainder of the chapter is devoted to the study of the
following important examples' of functions and capacities
which are alternating of order ': the operation sup' on a
group lattice; increasing valuation on a lattice (for example,
a non-negative Radon measure); functions derived from a
probabilistic scheme; exponentials; energy of the restriction
of a measure to a compact set; and others.
CHAPTER V I. -- (lapacitability. Fundamental theorems.
First, the alternating capacities are studied: we establish
conditions, to be imposed on E, g, and f, which will suffice
to insure the preservation of capacitability under finite or
denumerable union, and which will imply the validity of the
relation
f,(U An)--- lim f*(A)
where
It then follows from a general theorem that every K
contained in a Hausdorff space E is capacitable with respect to
every alternating capacity f defined on {(E). In order to
pass from, these K sets to arbitrary borelian and analytic
sets, we use the fact that every analytic subset of E is the
projection on E of a K contained in the product space
(E X F),' where F is an auxiliary compact space; and we asso-
ciate with [ the capacity g on ,,(E X F), where g is defined by
g(X) --/(/r,X).
It is then easily proved that every g-capacitable subset of
(E X F) has a projection on E which is/-capacitable. Now, g
is alternating as well as [; hence, every K' of (E X F) is
g-capacitable. From this follows the f-capacitability of all
analytic subsets of E.
A number of counter-examples show that it is impossible to
improve on the results obtained:in particular, by using a
result of Goedel we prove that it is not possibl. e to esta-
blish the capacitability of all complements of analytic sets.
After givin.g several. applications of th. csc results to measure
'theory, we investigate monotone capacities. Their study is
lO
136 , c,s.v COQJ
reduced to that Of alternating capacities by means of the
concept of conjugate capacity. From the general t'tleorems
obtained in this way, special cases such as the following ar.e
derived:
If E is a complete metric space, and if [ is a monotone
capacity of order 2 on .(E), (/(A B)q-/(A a B)f(A)q-/(B)),
then all borelian subsets of E and all complements of analytic
.sets are capacitable.
CHAPTER VII. -- Extremal elements of convex cones and inte-
õral representations. Applications. In this chapter, we study
several convex cones whose elements are .fUnctions; we deter-
mine their extremal elements and employ them to obtain inte-
gral representations of these functions. The basic tool for
these representations is the theorem of Krein and Milman
concerning convex and compact subsets of locally convex
spaces, and its immediate consequences. This theorem enables
us to state the existence of such a representation in the case
of a cone such that its base and also the set of extremal point
of the base are both compact. Uniqueness of this represen-
tation implies that the cone under consideration is a lattice;
but it has not been proved that this condition is sufficient to
insure uniqueness.
We study in this manner the positive increasing functions
defined' on an ordered set, the increasing valuations on a
distributive lattice, and, in particular, the simply additive
neasures defined on an algebra Of .sets; for these we use the
compact spaces which Stone associates with these algebras.
The study of the cone. of all positive functions which are
alternating of order infinity on an ordered semi-group S illus-
trates the significance of the exponentials , defined on a
semi-group (0 ___< + <__ _, and b(a-r b) ',' +(a). +(b)). When
S R. or S R theorems analogous to those of S. Bernstein
+]
are obtained; when S is an additive class of subsets of E, then
we find extremal elements each of which is characterized by
a filter on E.
In seeking a way to study 'the cones whose elements are
capacities on .';.(E) we are led to' the introduction of a½½ vague
topology on the set of all positive increasing functions œ
defined on ..(E): this is achieved by the use of the extension
of œ to the set Q+ of all numerical functions defined on E
which are non-ngative, continuous, and zero outside of a
compact set.
It is then proved, for instance, that if E is compact, the set
of all capacities œ which are positive and alternating of order
on ,.(E), and which satisfy the relation f(E) , is compact
in the vague topology, as is also the set of its extremal points.
This leads to a remarkable probabilistic interpretation of
these capacities, and makes it' possible to prove that the class
of these capacities is the least f'unctional class containing all
positive Radon measures, which is stable in a certain sense,
with respect to continuous mappings.'
Thereafter, we take up' the study of those classes of func-
tiondis on Q+ which may be obtained from the primitive
functions [, defined by the relation [.(9) 9(a) by means
of the following operations: superior envelope, inferior enve-
lope, and integration (g=
The chapter ends with the study of the relations between
the pseudo-norms defined on a vector lattice V and the func-
tions œ which are strongly sub-additive on V.
CHAPTER I
BORELIAN AND ANALYTIC SETS IN TOPOLOOICAL SPACES
1. Introduction. There , are difficulties in extending to
an arbitrary topological space E the classical results concer-
ning the parametric representation of borelian sets. For
example,. in a general setting each sub. set of E is the continuous
and -t mage of an open set of a suitable compact space; for
one can easily construct a compact space which contains an
open set of a. given cardinal and each of whose points is iso-
lated.
In order to obtain theorems of interest, one is, therefore,
led to modify the classical definitions. In particular, we shall
have to eliminate the open sets and begin with the compact
sets, which possess topological characteristics invariant under
continuous mappings. Therefore, we shall be led to replace
the sets Gs, whose role is fundamental in the study.of classical
borelian and analytic sets, by he sets K which we shall
define in terms of compact sets
1. t. Dr.tXTXOrq. A class ß of subsets of a set E g,,hich
contains the intersection and the union o[ any denumerable
[amily o[ elements o[ ß ,,ill be called a borelian field on E.
[. 2. DEFINITION. If E is a Hausdorf/ topological space,
the smallest borelian field on E mhich contains each compact set of
E mill be called the K-borelian field o[ E and denoted by $(K).
The members o[ $(K) ,ill be called K-borelian sets.
2. Classification of K-borelian sets. One can show, as in
the classical theory, that the K-borelian. field of E is the
increasing union of a transfinite sequence of type :). of classes
THEORY OF CAPACITIES :139
where
(i) .'0 designates the class of compact sets of E;
(ii) S designates the set .of denumerable i.nters.ections
(unions) of elements belonging to .. where < ff a s even
(odd), the limit numbers a being considered as even.
We shall designate in general the classes with finite indices
by , ,., .,a, ..., and we shall say, for example, that a set is
a K,a if it belongs to the class K.
2. . Immediate consequences.
(i) Each finite union or intersection of sets of one class.
belongs to that class. Each denumerable intersection (union)
of sets of .qf. belongs to .ff, if a is even (odd).
(ii) A simple argument by transfinite induction shows that
each K-borelian set of E is contained in a K, of E. It follows
that if E is a separable and complete metric space which is
nowhere locally compact, not all borelian subsets of E, in the
classical sense, are K-borelian; on the other hand, we shall see
later that all classical borelian subsets of a separable and
complete metric space are K-analytic.
If E is such that each open set G of E is a K, each closed
set F of E is, of course, also a K; then the field of borelian sets i n
the classical sense is identical with the K-borelian field ().
This is the case when, for example, E is a separable and locally
compact metric space.or, more generally, when E is a metric
space which is a K.
3. K-analytic sets. We shall now define a class of sets
logous to the classical analytic sets.
ana-
3. 1. DEFINITION. In a. Hausdorf[ topological space,
each subset g,,hich is the continuous image o[ a K, contained
in a compact space geill be called a K-analytic set.
3. 2. THEOREM. Each subset of a Hausdor# space hich
is the continuous image of a K-analytic set is also K-analytic.
The class of K-analytic subsets of a Hausdorl space is a bore-
lian field.
(6} It would be interesting to see, if, conversely, this identity entails that each
open set of E is a K.
140 usrxw caoqur
Proo[. The first part of the theorem follows immediately
from the transitivity.of continuity.
In order to establish the second part of the theorem, .let
A,, A, .. .., be a sequence of K-analytic sets of E,
where aa"isA[ image, under the conti.nuous mapping f,,
of the set B F,, where B s a K and F s a compact space.
Let us show first that A, U A is K-analytic. Let F
be the compact space obtained by the compactification of
the topological sum-space EF by the addition of the point
at infinity; let B--UB; the set B is by definition a subset
of F.
We shall designate by [ the mapping of B into E whose
restr!ction to B is identical to f,; this mapping is clearly
continuous and we have A ". œ(B).
It remains only to show that B is a Ka. Now by definition
we can set B- B,, (i' 1, 2,...,) where each B, is
ß a K of F. Since the F of F are mutually disjoint we have
B-- B-' (B.). Since V B, is a K, B is indeed
a K.
Finally, let us show that A--(' A is' K-analytic. Let
F - 1-[ F, the product space of the F, and designate by C
the subset of F defined by C "]_IB. The sel C is the inter-
section sof a the cylinders b of F wh. ere b--.X I-,F;
each b i Ka, and therefore the same s true of
Furthermore, we shall designate by [ the canonical exten-
sion 'to ba of the given 'mapping of B into. E; [a is theref.ore
defined at each point of C. The set of points of C at Which
[ :... [ is closed relative to C, for each n, since [, and f are
continuous; therefore the set of points of C at which ::/
for all i and ] is the intersection of C and a closed subset of F.
This intersection is therefore a Ka which we Shall designate
by B.
W.e s.hall. design. ate by f the restriction of the f to B. This
restnchon s conhnuous on B and since/(B) c A for every n,
we have /(B)A. On th.e other hand A/(B). For let
y A; for every n there exists an x B such that f(x,,) y.'
THEORY OF CAPACITIES
The point x.. (x,) of F belongs to B and we have therefore
(x)- y. Thus A./(B) and A is the continuous image
f a K,.
3. 3. $oulin's operation A. .-Suppose that Ax is a class
of subsets of a set E where ), denotes a finite sequence
(n,, n, ..., n) of positive integers. For every infinite sequence
s- (n,, n, ..., n, ...) of positive integers, set
The set A-- A, is called the nucleus associated with the
$
class I A,I; it !s,also refe. rred to as the
Class by Soush n s operation.
set obtained from this
Let S denote the topological space of all sequences s, lexi-
cographically ordered with the. topology induced by that
order; then it can be easily shown that A is the canonical pro-
jection on E of a set .(E x S), with A,- &, where
i
each Ao(i- , 2, ...) is a Countable union of elementary sets of
the form (A X x), with x denoting an interval of S. An
immediate consequence of this is the following theorem:
Tsoas. If a subset of a Hausdorf[ space E is obtained
by Souslin's operation [rom t class [ K-analytic sets, then that
subset itseli is K-analytic.
DEFINITION. Every subset of a Hausdorf[ space E, obtained
by Soulin's operation [roma class o[ compact subsets o[ E is
called a K-Souslin set.
It is easily shown that, if [ is a continuous mapping from a
compact space E into a Hausdorff space F and if B ½ F, then
the set A /-'(B) is a K-borelian set of class K (respectively,
K-Souslin, K-analytic), if B is of class K (respectively K-Sous-
lin, K-analytic).
3. 4. D:XT. XOS. ..A subset A of a Hausdorf[ sp.ace is
called a set of uniqueness A is the continuous an d 1-1 mage
o[ a K o[ a compact space.
3. 5. T.oazM. E,ery denumerable intersection of sets of
uniqueness is a set o[ uniqueness. E,ery denumerable union
of disjoint sets of uniqueness is a set o[ uniqueness.
i42 GUSTAVE CHOQUET
Prooœ. For the first part we may refer to the end of the
proof of Theorem 3. 2 and remark that if the œ are l-l, then
there exists in B a single point w--(xa) such that [(). y.
The same remark applies to 'the second part.
4. The K-borelian sets. Later in this work we shall use
the fact that the K-borelian sets are K-analytic. More pre-
cisely, the following theorem holds.
ß . l/ ß
4. i. Toae. .. E,e. ry K-borelan set s K-ana tc.
Furthermore, i[ the Hausdor# space E has the property that
each subset o[ the [orm KG is a K, (where G is open), then
each K-borelia n subset o[ E is a set o[ uniqueness.
Proo[. The first part is an immediate consequence of the
fact that each compact set is K-analytic. The field of K-bore-
lian sets is therefore a subfield of the field of the K-analytic
sets.
We shall prove now the second part of the theorem. Assume
at first that E is compact. Then each open set G of K is a
K by hypothesis. The borelian field generated by the open
sets of E is identical with the increasing union of a transfinite
sequence of 'type O. of classes
..... .
where
ordinals a.
We shall designate the classes with finite indices by
q, (, q, .... Since each G is a K we have (0
Likewise, by taking complements, we have :h' 0 ($,. By trans-
finite induction it ollows that for each a < we have (
and :ff(.,+,. Moreover, is identical with the set of
complements of elements of ..
Let us suppose then that for an even a we have shown that
each element of ( and of / is a set of uniqueness; the
same' is true of the elements of (.,, because the class of sets of
uniqueness is closed under the operation of denumerable
intersection. Then. let I{ be an element offs.,. Bydeft-
(i) (90 denotes the set of open sets of E;
ii ($ denotes the set of denumerable unions (intersec-
.() .
tons) of elements .belonging to the ( where < if . !s
even (odd) with the same convention as above for the hmt
THEORY OF CAPACITIES 1.43
U
nition we have K,+, K, where K, a and we can
always suppose that the K, form an increasing sequence.
We have
K' fq K .... Ki 'fqG. This set is the nter-
K,+t,. K
section of two 'elexnents of (+; hence it is a set of uniqueness.
Therefore K+, which is a denumerable union of disjoint sets
of uniqueness, is a set of uniqueness.
It can be shown similarly, by interchanging the roles of
and 3 that if, for odd, the elements of and 3. are
sets of uniqueness, the sane is true of the elements of
and 3i:. .
Now if a is a limit number (and therefore even), and if for
each < a the elements of ! and of are sets of uniqueness,
the same is true of the elements of (g and of ;)i'.
This is obvious with regard to 3 since it is true of denume-
table intersections; for { this follows from the fact that
each (g can be written in the form of a denumerable union of
disjoint elements of classes with . < . By transfinite
induction each element of a ( (or 3.) is therefore a set of
uniqueness.
Consider now the ease where E is not necessarily compact.
If A is a K-borelian set of E, it is contained in a K,
where the Ka are compact and increasing with n. Therefore A
is the union ofthesets Alq K, and Aft(K,,+,- K) for n .-- 1, 2,...
Each of these sets is a K-borelian set and is contained in a
compact set. They are therefore sets of uniqueness. Since
they are disjoint their union is again a set of uniqueness.
4. 2. RztAalc. If E is a separable conplete rnetric space,
we have already observed that a subset of E which is borelian
in the classical sense is not necessarily K-borelian. On 'the
other hand, since such a space is honeonorphic 'to a G, of a
compact metric space, we can assert that a subset of E which
is borelian in 'the classical sense is homeomorphic to a K-bore,
lian set. Such a set is therefore always K-analytic. This
remark will allow 'us to apply our theory of capacities to the
subsets of separable complete metric spaces which are borelian
or analytic in the classical sense.
144
GUSTAVE CHOQUET
4. 3. REMARK. We have not obtained in the prece-
ding .all the results parallel to those concerning the borelian
sets n the classical sense. We shall state here a few of
these in the form of problems.
PROBLEM. If a subset A of a compact space E is
homeomorphic to a K-borelian set of class .%: (respectively
K-Souslin), is A a K-borelian set, and if it is, is A of the class
(respectively K-Souslin) ?
¾
The results of Sneider {1 and 2] (') show that the answer to this
question is affirmative whenever E is such that the union of
every class of open subsets of E is union of a countable 'subclass
of that class of open subsets.
4. 5. PROBLEM. If E is compact, is each subset of uni-
queness (or more generally, each continuous image (0 i)()
of a K, of a compact space F) a K-borelian set?
4. 6. PROBLEM. If E is compact, is every K-analytic
subset of E also a K-Souslin set?
5. The operation of projection. In the classical theory of
analytic sets one shows that each analytic subset of a Eucli-
dian space R is the orthogonal projection of some G of a
space R "+' containing R . We shall need later the following
analogous theorem.
5. t. Tnsoas.
If E is a Hausdorf[ space, then each
K-borelian subset o[ E (and more generally each K-analytic set
hich is a subset of a Ko) is the canonical projection on E o[
a Ka o[ the product space o[ E and a compact auxiliary space.
Proo[. The proof will be given first under the assumption
that E is compact. If the set AE is the continuous image
under the mapping f of a set B, which is a K in the compact
space F, the set A is the projection on E of the graph F (E X F)
of the function y -- œ(x) defined on B.
Now the continuity of f implies that [' is identical with the
intersection of [i and the product E X B, that is to say,. [' is the
(*) Numbers in square brackets refer to the bibliography given at the end of this
report.
() An application is (R,.,-- I I if the inverse image of every point is at most enu-
merable.
TIIEORY OF CAPACITIES
intersection of a coXnpact set ;vith a K. Therefore F is a
K,, which proves the theoren.
More generally, if E is a Hausdorff space and if A is K-ana-
lytic and contained in the union. .JK,, of compact sets K of E,
then A is the continuous image, by means of the function f, of
soxne B, which is a K contained in the stun jZF,, of compact
spaces F,, such that /(BIqF,,)cK. If we take for F the
compact Space obtained by the Alexandroff compactification
of LF, then the graph of [in E X F is stfil a K, and its
projection on E is identical with A.
CHAPTER II.
NEWTONIAN AND (iREENIAN CAPACITIES
6. Newtonian and 6reenian capacities. Let D be a
domain in R which possesses a Green's function. (For v > 3
any domain D possesses a Green s function, but for v "2 there
are domains D which are not , Greenian ).
Let G(P, Q) be this Green's function, and let be a Radon
measure on a compact subset KcD. The potential of , for
ß this kernel G(P, Q) is by definition
U,(Q)- f G(P,
If Ix is positive, this potential is positive and superharmonic
on D; it is harmonic on (D K) and tends to 0 whenever Q tends
toward a point on the boundary of D, with the exception of
the so-called irregular frontier points, which fomn a rare-set
for example,
in a sense defined in modern potential theory (see,
M. Brelot [l]).
Let us say that a positive measure , on K is
. i everywhere D. The total mass
admissible if
U(Q) .. on of is the
integral fdlx. The supremum of the total masses of admis-
Sible measures on K is called the capacity of K (relative to D).
For example,. t. he capacity of K is zero if the potential of each
non-zero positive measure On K is unbounded on D.
For a fixed 'domain D, this capacity is denoted by f(K).
The following properties of œ(K) are well known.
f(K) > 0 and is an increasing function of K, that is,
f(K,) f(K)
THEORY OF CAPACITIES
147
6. 2.
f(K) is subadditive, that is,
f(K, U K) f(K,) q- f(Kd.
For let , be an admissible measure on (K, U K) whose tofal
mass m differs from/(K, U K.) by less than ,. If , and _ are
the restrictions of I to K, and K,., respectively, and if m, and
m are their total masses, then m . m, q- m_ and and ,. are
admissible. Then m < m, q- m, _ [(K,) q- [(K), and the
inequality stated above follows.
We shall soon see, in fact, that/(K) satisfies muc. h sharper
inequalities which, in a certain sense, cannot be mproved.
6. 3. f(K) is continuous on the right.
This means that for any compact set K and any number
a > 0, there exists a neighborhood V of K such that for every
compact set K' satisfying the relation K K' V, we have
0<f(K') f(K) _<,. The proof of this property will be
omitted.
6. 4. Interior and exterior capacities. (lapacitability, . We
shall associate with every subset A of D an interior capacity
and an exterior capacity.
We define the intergot capacity of A to be sup f(K) for K A
and denote it by f,(A). In particular, the interior capacity
of every open set G D is defined. This fact enables us, then, to
defin.e the exte.rior capacity of A to be inf/.(G) for A c G; the
exterior capacity of'A is denoted by/*(A). Thus, for every
open set G we have f,(G) if(G). More generally we shall
say that the set A is capacitable.if L(A) if(A), and we shall
designate the common value of the two capacities by œ(A); the
notation f(A) will not lead to confusion since, as will be shown
later in the general theory of capacities, f,(A)- if(A) f(A)
whenever A is a compact set. (This result follows easily from
the continuity on 'the right of [.)
We say that a property holds quasi e,erywhere (nearly
eerywhere) if it holds at each point of D except at the points
of a set of exterior capacity (interior capacity) zero.
When the set of exceptional points is capacitable, the two
notions coincide; we shall see in the following chapters that
this situation occurs ;vhen the set of exceptional points is
borelian or analytic.
t48
GUSTAVE CHOQUET
We now prove the following property, which will soon be nee-
ded: The union of a finite number of sets of exterior capacity
zero is a set of exterior capacity zero.
For if if(A,)-/*(A) 0, there exists, for every
open .sets G,-and G containing A, and A., respectively whose.
capacmes are less than . But [(G, U G)f(G,) q- f(G,.)< 2a,
indeed,-each compact set K contained in G, U G is the union
of two compact sets K, and K such that K,G, and'
K G (). Then/(K)_</(K,).-F f(K)/(G,)-4- f(G.) since
/(G, !J G.,.) /(K) can be made arbitrarily small, the ;subad-
ditivity for open sets follows.
arbitrarily small, we have
complete.
Since f(G, U G.) can
f(A, U A.) 0. The
be Inade
proof s
Equilib ' ß
6. 5, rium potential. It is shown in potential
theory that for every compact set K c. D there exists one and
only one admissible measure , defined on K such that its
potential U is quasi everywhere in K equal to 1. Its total
mass is equal to the capacity/(K) of K. This measure is the
equilibrium distribution of K and its potential is the equilibrium
potential of K. The equilibrium distribution is the only
admissible measure on K whose total mass is equal to/(K).
6. 6..Fundamental principles. We recall the following
two assertions which we shall need presently.
Let U be the potential of a Hadon measure. defined on a
compact set K D and such that U is bounded on D.
6. 7. If U _. 0 quasi e,erywhere on K, then the same inequa-
lity holds e,erywhere on D.
The property stated in 6. 7 is an immediate consequence of
the general maximum principle. We shall not state this
principle however, because it involves the notion of energy
which we shall not use.
6. 8. If U, 0 e,erywhere on D, then the total mass of is
positi,e; it is zero only when UO.
It follows readily from these two 'properties that, if U_ 0
quasi everywhere on K,. then the total mass of is positive;
it is zero only when U,0.
{} For a proof of this fact see 17.4, Chapter v.
THEORY OF CA. PACITIES
t49
7. Successive differences. If ?(x) is a real function of
the real variable x0 the fact that ? if increasing may be
expressed by stating that A,(x, a)- ?(x-k-a)--?(x)_O for
all a > 0. Similarly, the fact that it is convex may be expres-
sed by stating that
"', 9(x -4- a q- b) T(x + a:)- ß 9(x -4- b) -4- 9(x)>.0
for all a, b >_.0. More generally, if ? has a derivative of order n
and if this derivative has constant sign, then this fact may be
stated by' saying that the difference A, of order n always has
this same sign.
The successive differences of ? then furnish a means of
studying the nature of the increase of . This method is of
interest because it can be extended to the study of functions
not necessarily of a number x, but of a set, or more generally
of elements of a commutative semi-group, addition being
replaced by the. semi-group operation.
It will be shown presently that the successive differences
relative to the capacities œ(K) are alternately positive and
negative; therefore, it will be convenient to so alter the sign
that the final expressions all have the same sign.
.capacity of K. If X, A,, A, ...,
we define
V, (x; A,),, h(X)-
anti, in general,
7. 1. Successive differences relative to equilibrium potentials
and to capacities.- For every compact set K D we desi-
gnate by h(K) the equilibrium potential of K, and by/(K) the
are compact subsets of D,
h(X U A,)
A,,+,),,,, V,(x; A,,
V.(x U A,,.,.,; A,,
The differences V,,(X; A,, ..., A,)y are defined in the
way.
The index [ or h will be omitted when no ambiguity
possible.
same
Functional properties of the differences V,,-
7. 2. (X; A,,..., A) is a symmetric function of the
variables A. This property is a consequence of the following
development of V.:
+ (., u A, u
UA.).
..This symmetry permits V.. to be written in the form
V,(X; !A, t). The index n may as well be omitted since it is
determzned when I is determined.
the family A
7. 3. V,(X; IA, I)--' V,(x; !*;I) if x u A, XU A'
for all i. This follows from the fact that the A, always occur
in the development of V in a union with X. In particular,'
V.. = 0 if A X for all i.
7. 4. V.(x; IA, I) ..-- V.(½; IA, I)... V..,(½; IA:, xl)...
where the expression A, X I denotes the family oz sets consis-
ting of X and the A. This formula is easily derived from
the expression that defines V,.,(; .A,, X I)in terms of the V,.
It' shows that V, is the sum of two functions, each of which
is a function symmetric in all its variables.
7. 5. V(X; a,,..., A,_,,.A U a).. V,(x; A,,..., A,)
.. V.(X U A.; A,,..., A._,, a..
In order to verify this relation it is sufficient to express each
of the V.. in. terms of V,,_,. The six terms thus obtained
cancel pmrwse.
Fundamental properties of h(X) and f(X).
7. 6. THEOREM.
(ii For even d X and I A, i it is true that O.<--V,(x ; I A, 1 ),..-< 'l.
THEORY OF CAPACITIES .5.
.potent. jul ] V to 0 on
and it s an increasing [uncaon of each of the A.
(ii) This potential is a decreasing [unction of X, and more-
o,er it is a decreasing function of n in the sense that
V(X: i A,.,,)__.< -- V(X; {A,l,o) hene½,er
IzJ.
Proof. This theorem is proved by induction on n and by
using the functional properties of the V,,. To simplify the
notauon, let V,, ..
Consider first
is the potential of a measure defined on XUA, since
X .
V( ,. A,) -- h(X.U A,) .... h(X) Now 0.<___ h(X A,) < and
0 <__ h(X) < i, wth h(X) -- quasi everywhere on X and
h(X U A,) .' .'. i quasi everywhere on (X U A,).
Th. us V', < ; V', .... 0 quasi everywhere on X and V' 0
quas ever-here 'on A,. "
Hence V', 0 quasi everywhere on X U A,; and, by virtue
of the fundamental principle 6. 7, V.', 0 everywhere. Moreover,
Vi(x; a,)is an increasing function of A,. This fact is an
immediate consequence of the functional property 7. 5:
V.', (x; A, Ua)
V'(X; A.,)--V'(x u A,; a)>0.
Consider next (i) in the general case. We suppose the first
part of the theorem to be true for all p < n and show that it is
true for p -- n + . Since
V' )
+.,(X; A,, ...,
V'(X; A,,
, A) V'(X
... U A,; A,, ..., A,,),
is the (bounded) potenual of a measure defined on
For each ,V' of the second member, 0 < V' ! everywhere,
so that V+,.< everywhere. Each of the V is zero quasi
everywhere on X, and therefore similarly for V' On A ,
n-- I -
11
52 vsxv. cnoqv.
the first V., ,is .positive. and the second is quasi everywhere
zero; thus V.. s quasx everywhere positive there. Because
of the symmetry of V.+, with respect to the variables A, the
preceding result holds for all A. Then; since the potential
V'.., is quasi everywhere positive on the union of X and the A,
... 2> 0 everywhere. Also, we have on X:
V'(x;&, , A.).
,. ... A,,
ß 0 0 .-= 0 quasi everywhere on X.
This completes the proof of our assertion that ( V,), is
equal to 0 quasi everywhere on X for every n.
That V'.+, is an increasing function of each of the A is an
immediate consequence of property 7. 5, just as in the case
of V
Consider next the proof of (ii). Clearly
V'(XUa; {A,J)...V'.(x; {A,J) V'+,(x;
which shows that V' .
ß is a decreasing function of X
From this same relation we see that
A,J) .<__. O,
Ix, l),
V'
that is, decreases whenever an element is adjoined to the
family of the A; therefore whenever any number of elements
' djoined
1S a .
Complement of theorem 7. 6.
7.7. D.rNTION. The essential envelope [4 of a compact
set K D is the closure of the set of points of D on which h(K) I.
The. set I is compact and (K ... Rt) is~ a set of exterior
capac, ty zero; the relationship of K o K is expressed by
saying that K is quasi contained in <. Since h(K) h(I),
we have I- [4. Similarly, (I, .) implies I7(, c ; and,
moreover. K, U K
(K,U f<) for any choice of K., and K.
THEORY OF CAPACITIES
i53
7. 8. Restrictive hypothesis'on --We suppose that for
all K D the open Set (D -K)?s' connected; this will be tt/e
case if the frontier of D is connected (this frontier contains
;he point at infinity if the dimension of D is greater than 2
and if D is 'unbounded). -
When the condition (D K) connected for e½,ery K D,
is satisfied, s,e g,,ill say that D is simple.
7. 9. Statement of the complement of theorem 7. 6.
When D is simple, a necessary and sufficient condition that
(X; A i ) 0 on D is that there exists an o such that
.X, ø . When V(X; I A, I ) -5 & 0, the set of points of D gehere
V- 0 is contained in and differs [rom k. by a set of exterior
capacity zero.
Proo[. We shall use 'the following fact:if h[3 and if
A--/:B, then at every point of lift we have ~h(A)? h(B).
For let m a l Jr. There exists a po. int m0a(B_..A] s. nch
that all' spheres S with center m0 ntersect '(B A) n a
compact set b of non-zero capacity.
small so
a[k g] .....
We have, a fortiori, h(l))
Consider first the case
is identically zero if
If S is taken sufficiently
.that D (f3 LI/) is connected and m ß b, then
.... h[/,] is harmonic and strictly positive on I ( O )
V,(x;
which is equivalent
h(X LI A)
to
Jt C ; otherwise V, 5/= 0 at each point of l (. Moreover, we
.know that' ¾, .. 0 quasi everywhere on X.
Consider next the general case. We now assume the asser-
tion true for p n and prove that it holds also for p -- n q- I.
If one of the A is such that /0 , then V.,,,--0. Other-
wise, consider
A,,
ß e e
V!
.. - (x.u ;A,, ...,
The first term of the difference is greater than 0 on
.54 GUSTAVF. CHOUF.T
and the second term of the difference is zero quasi everywhere;
then the difference is greater than 0 quasi everywhere on that
set. At every point of I (([.]A.)UX)), the difference
is positive and harmonic; it is therefore greater than 0. Thus
V'., is quasi everywhere greater than 0 on I'.
In fact,
The proof
this strict inequality holds everywhere on I .
is entirely analogous to the above. We replace each with
U b], Where each b is compact and small enough so that we
may conclude that a certain harmonic function is greater
than 0. Finally, the theorem follows from the fact that, as
we have already seen, V'
, 0 .quasi everywhere on X.
7. 10. Co o TXSo ash 7.6. If (¾,)ydesignates the
dif[erences associated with the caps. city [, ve.ha,e (V).r< 0 and
(V)ypossesses the same moaotomc properties as
Proof. The potential (V,,.)ais a linear combination ofpoten-
rials h(k), and the total mass of the measure which generates
it is the sum of the total masses of the equilibrium distribu-
tions of the compact sets K, with the same coefficients, q- l
or i, as the corresponding potentials h(K). Moreover,
according to the second fundamental property 6. $ of poten-
rials, since (Va)a < 0 everywhere, the total mass of the measure
which generates' it is negative. Thus (/a)y:< 0.
The monotony properties of (Va)y follow, as in the case of
the (Vn) , from the functional prope. rties of the V,, and from
the fact that all the (Va)y are negat,ve.
7. [. COMPLEMENT OF COROLLARY 7. 0. We deduce imme-
diatelt [rom the complement 7. 9 o[ Theorem 7. 6 that, under
the h!ipothesis that D is simple, a necessart and sufficient conditio n
that V (X; {A l)- 0 is that for some i i0, ve ha,e ,o ).
7: 12. Ra.xaK - Whenever a function ?(x) of a real
variable x satisfies inequalities analogous to those shown for
TIt CORY OF CAPACITIES [55
the capacity [, it is increasing, concave,... and possesses
derivatives of all orders, alternately positive and negative.
The opposite--q of such a func.tion is said }o be comp. let. ely
monotonous although the term s not especially descnpt¾e.
It is-known that such a function is analytic. The capacity
thus appears as an analytic set function, with deri,ati½,es
alternately positive and negative. We shall say that a capa-
city is a set function which is alternating o[ infinite order.
8. The inequality (V,)y 0. - This inequality
written as follows:
can be
s.t. f(x) -f(x U A,)
, f(x U A) + f(X U A, U'Ad O.
IrA and B are any two compact sets, let X ..... ' A fB, A,
and A- B. Then the inequality 8. i implies
'A
8. 2o
f(A U B) + f(A FI B) f(A) + f(B).
Since [... 0, the capacity satisfies an inequality stronger than
subadditivity. This inequality plays an important role in
the following.
We .remark here that ordinary subadditivity is sonetines
wrongly called convexity. In fact, the preceding inequality,
which is stronger than subadditivity, is, as we' have seen above,
analogous. to a condition of concavity.
We shall proceed to give another form to the condition
V. 0. Let a, k, A, be three coxnpact sets with a c A.
Setting X a, A, "k, A.,. A, it follows that
8. 3. f(A U k)--f(A)<f(, u )- f(,:,,).
In other words, when a fixed compact set, k, is adjoined to a
compact set X, the smaller the X, the greater the increase in
the capacity of X.
APPLICATION. Let (A) and (a,), i :'. 1, 2,..., n, be two
[amilies o[ compact sets such that aC Ai [or all i. Then
8. 4. f(L.J A,)-..f(L. Ja,)Y,(/(A,) f(a,)).
156
GUSTAVE CHOQUET
Proof.
vritc
According to the inequality 8. 3. above, we may
f(A, U Aa)--f(a, U A.)f(A,) f(a,),
f(A U a,)--f(a, U a.,)_____< f(A..,). f(a),
from which. adding termwise;
8. 5. f(A, U A)---f(a, U a)
'? (f(A,) f(,,)).
/_z_
i-" t,2
If the inequality 8. 4. is satisfied for order n, it is sufficient
to apply inequality 8. 5. to obtain 8. 4. also for order n q- 1.
8, 6. Geometric application of the inequality V,..<: 0.
We shall suppose D to be invariant'under a one parameter
continuous group of motions T., vhere the parameter ), is
chosen so that TL.T ...' T,, +.L.. For every compact set Ko
in D and all pairs of values a, , of X let K:, b -- J Tx(K0)
and K b . K0b.
In other words, K, is generated by the
varying between . and .
motions of K0 for ),
Because of the invariance of D vith respect to the
clearly [(K,) ' [(K(i_)). Let f(Kb) -- ?() and
Ko,, A,; K,,,o:, +,,,) -- X;
A2, where a,, -2, x
are positive numbers.
Then the inequality V._, 0 becomes
+ ß + + + ,., + o.
Hence, the second differences relative to ?(x) are negative,
that is, ?(x) is a concave function.
Thus the capacity o[ K is an increasing conca,e [unction o[ .
This property can be easily verified for the solids of R
whose capacity can be explicitly calculated.
EXAMPLE.- If the T represent translations in D ' R
the K are unions of parallel segments of length .
9. Complete system of inequalities. We have obtained a
system of inequalities V,.0 which are satisfied by the
function f(K). We shall show that in a certain natural sense
THEORY OF C&PACITIES
157
there are no others, that. is, that every inequality identically
satisfied by f is a consequence of V.,0.
Let i AIs, (I-= t l, 2, ..., n J) be a family of n compact
subsets of D. For each J c I, let
aj
and xj ,-- f(B,) for J
There are N 2 " ! subsets J of I. We may then associate
with each family IA*Ie the point of the Euclidean space R s,
whose coordinates are (xj)jct. Our object is to characte-
rize the locus of this point in R s when D and I remain fixed
but the family l Aila[ is allowed to vary.
9. t. DEFINITION. We denote by C. the set consisting of
the points (xj)jct of R when the family I A,],a ,aries, land D'
remaining fixed. We denote by L. the set oonsisting of the points
of R defined by the follo,ving N inequalities:
V (B,_H;
We have omitted,
I AJ,,,) < 0 ,here H I and H -% .
in this definition, the index of V which is
obviously equal to H.
The second part of this definition requires an explanation.
Each V is a linear combination (with coefficients q- ! or )
of terms of the for,n /(B); if we then replace each/(B) kY
x we have a form which is linear with respect to the x. The
set of points of R for which V.<0 is then a closed half-space
in R. s. More explicitly,
where
(I--H).
9. 2. THEOREM. (i)
dimension N; it can be represented parametrically in the [orm
The set L is a con½,ex cone of
x.q. (x.>0)
mhere the ,ector V of R
has the components x}" defined as folloges :
0 if HfqJ--. % and x] 1 if
(ii) C,, cL and C L,,().
Hf3J :.0.
(s) The notation ]k means the interior of A.
i58 GUSTAVE CIIOQUET
9. 3.. Proof of (i). We shall use the expression of ,,, X
a function of the x obtained above and calculate
as
for a J0 c I.
The coefficient of x.o is
where (I H) Joand
H
H CI J0-= .
It follows that this coefficient is
i.
Similarly the cceiticient of xj for J J,, is
where (I - H) J and
H f'l Jo =/= ½.
By examining first the case where J0 J and then the case
where J,, ½ J, we find that the coefficient of x. is always 0. Thus,
H/'lJo--o
This gives the solution of the system of equations
JDI--H
The second members of these equations are thus linearly
independent forms, and the vectors V. are also linearly inde-
pendent.
The formula OM E),aVa follows immediately from the
expression of the xj as functions of the
9. 4. Proof of (ii). The relation C,, L, is an immediate
consequence of the fact that, for every point of C,, the /
associated with this point are all negative, according to
corollary 7. 10. The relation C,, L,,, which expresses the
identity of the interior of C, and the interior of the cone L,,, is
much less obvious.
We present here a general outline of the proof. Let us suppose
for a while that for every system of numbers k.0 (with
ß oa¾ or cr^cm.s i59
H 'I and H =/= ) there exists a family of compact sets Ka
with /(Ka) .... ka, which are additive in the sense that for
every subfamily I K.ot of this family, we have
f(O K.o)¾ ,
-- ./.f(K.,) ..... Y;%.
.,,..i
For this family of compact sets and for each i a I, let
A--L.J K.. In the space R "x, the point M representative
H)i
of the system of sets A, is then defined by
For we have here, with the
f(Bj) ,.-:-, f(ig ai). Now
notation already introduced,
f(gA,)=f( U
ß ' It f'lJ :-o ,
We have
then, x., ' N, TM ),.. Thus, under the initial
fIN J
.hypothesis of additivity we see that. every point of the cone L,,
is a point of C,,.
As a matter of fact, this hypothesis is realized only approxi-
mately, in a sense which we shall make precise, for the capa-
cities considered here.
'We shall use a hypothesis .a little different, and, in fact, weaker
than that of additivity, and attempt to show that it is realized
for o'ur capacities.
We shall suppose that for any given number s ,'> 0, there
exists a family of/compact sets K. c D (H c I and H =- p)
such that
9. 5.
for each of these we have f(K.). 1;
9. 6. f(UK,,) _ Zf(K.)
9. 7. for every X such that 0=),1 and for every H,
there exists]T,aco,npact K.(),)such that
(,:,)
f(K,,(),.)) ),,,
K,, (X') c K.(X)
K,-,(I) K..
if X' < X,
60 avsv
For every system of numbers k__0 such that }.])m< and
for every i I, let
A, = [,..j
We designate by m the point of L,, defined by
The set of these points, 'under the condition
is a simplex S o dimension N. The definitions above of A
associate with each m the point M .... (m) representing in R s
the family of the A. If the K.i(X) formed an additive
family, the mapping would be an identity. 'We shall see
,that with our hypothesis, is a continuous mapping which
differs arbitrarily little from an identity if is taken suffi-
ciently small.
9. 8. is continuous. It is sutlicient to show that
each f(B) is a uniformly continuous function of m; since f(B)
is an increasing function of Xa, it is enough to give to the
positive increments A),.. From the inequality
f(U U =< Z
it follows, since the K.(),) increase with ),, that
where the B and B are associated respectively with the
points m (ka) and m,+ Am (),a-4-AXa). This inequality
proves the required continuity.
9. 9. differs arbitrarily little from an identity. It is
sufficient 'to' show that each f(Bj) differs arbitrarily little from
Z )'a' More generally, given any family (Kp)ae of compact
sets such tha L f(UKp)< , the same inequality
holds when we replace each K by a compact subset of K r
TMs follows from the inequality used above by writing it in
'roY o cAxcs 16
the form f(U where
k K.
Now it.is a-well known fact that, if M -/(m) is a continuous
mapping of the N dimensional simplex $ of R" into R" such
'tha. t Mm.<'yi for all m, the image (S) of S contains all
points of S at a distance >'qifrom the boundary of S. Since 'r
ten. ds to 0 with a, it follows that each interior point of S is a
point Which represents a family (A) of compact subsets of D.
Finally, if we notice that in our second hypothesis the
constant which occurs in the definition of S, that is, in the
condition E)'a< i, can be replaced by an .arbitrary positive
constant a, we get immediately C, .-=. L.
.9. t0. Proof of the second.hypothesis.
hypothesis for the case in ,'which the
value 1.
It is sufficient to show that for every integer N and for
every 0 there. exists a family of compact regular .sets
.. 2, ... N), such that f(K) 1 for every i and
iK, (i 1, ,
We shall prove this
constant a has the
of cubes. For if C is a cube and if C½, denotes the cube
concentric with C and obtained from C by a homothety of
ratio 0, then /(C½,,) is a continuous increasing function
of ?. More generally, let K . .J C,, where each C,,'is a cube
and let K½, .'.J C,. Then, recalling the inequality (8. 4), it
follows that /(K½.) is an increasing and continuous function
of p with f(K½0)- .0 and f(K(,.) 1. The third par.y of the
second hypothesis s thus satisfied whenever the compact
sets K are regular.
Let G(P0, Q) be the Green's function for D with 'pole P0-
If S(P,,,' ?) denotes the open. Green's sphere defined by
G(P0, Q) ---?, t s well known that its capacity 1s --'
The procedure will now b e as follows; we shall suppose the
N points P, i--... 1, 2,..., N, so chosen that the restriction
of G(P, Q) to S(P, /2) is ..< for all couples i, / with i =f-i
( will be determined later as a function of a). Since for each
./D( i s K,) N - 'i, With q .<.'tl < a, wh.ere a compact subset of
ß called regular when t s the umon of a finite number
i62 GUSTAVE CHOQUET
i we have f($(p,, 1/2)) 2, we can find a regular compact set
of capacity > 3/2 in the open set S(P, 1/2). Starting with
this compact set, we can construct a compact regular subset
K of S(P, 1/2) with capacity ! by a procedure already
used.
Now, the equilibrium potential h(K) satisfies the relation
h(K) inf[l, 2G(P, a)] everywhere on D since
inf [.1, 2G(P,, Q)]
is 'the equilibrium potential of S (P,, ./2) and K, c- SKi.P,,
For every pair i, ] with i], therestriction of h(K)to s
then h(K)is, on each K,, less than (! -3- 28N). Then E h(K')
! q- 2;N
is on D the potential of a positive admissible measure (see
N
the beginningof this chapter) of total mass ß Thus
1 + 28N
N
l + nan =<f([-jK') _<N; hence, N--/( K){
+ N' For
given s and N, can always be chosen small enough so that
this quantity does not exceed s.
It remains to choose, for every 8>0, the N points P so
that the restrictions described above are satisfied. When D
has a boundary D which is sufficiently regular, we designate'
by 'z of. N distinct points of D.* and by
{ I a,.f.am. ily
mutually resjoint neighborhoods of these poxnts. Fo r every
i there exists a neighborhood W of ,': such that for
P a W and every Q a V, G(P, Q)(.. 8. If
then S(P, 1/2) V. Thus G(P, Q) ..:.< $3
i .-% i.
everv
" /9.
moreover < l ,
on S(P., 1/2)or
In the general case, a proof has been kindly iven by
M. Brelot () at my request.
9. i i. Study of the frontier of C.- .... We have proved the
relations C,, L., and ,,,, but it remains' to determine
which frontier points of the cone Ln belong to C.. This depends
essentially on the topological nature of D and probably on its
homology group. We shall give a complete determination of
(") M. Brelot. Existence theorem of n capacities, in these Annals, tome 5.
THEORY OF CAPACITIES
i63
C, only when D is simple (see 7.8). We shall not give a proof
here; it would be analogous to the proof of the second part of
Theorem 9. 2. and would follow essentially of the result stated
n 7. li.
We shall call the set of points of L,, defined by a set of
equalities of the form (Xs' 0) a [ace of L.. If this set
contains r equalities, the dimension of. the face is N -r.
The first essential fact is that if a point interior to a face of
L, belongs to C,,, then each point interior to this face also
belongs to. C,. Such an open face will be called an open [ace of C,.
9. 12. Determination of the open faces of C,,.
RULZ. Let Xs . Of be the set of equalities ,hich determine
a [ace o[L,. Its interior is an open [ace o[ C,i i[ and only
ß if I X,, . O}.is heredi.tary in the follo,in.g sense.: i[ it contains
an equahty ^ O, ,t must also contmn all ,ts descendents
relati,e to at least one index i o H.
For a better understanding of this rule, recall that we had
set Xa '(B,_.; l Al,aa) for any system of sets (&).
With th e hypotheses made on D, if .tXtat 0, there exists an
H such that A,o ½ B_. It follows the equality Xa--0
9. 13. EXAeLE.
The open face X.,,= "
The open face
implies that ks,- 0 for every H' such that. i 0 s H' and H' c H.
It is this fact that leads to the definition and the preceding
result. More precisely, a set g of equalities Xa 0 defines
an open face of L, if for every p there exists an i H such that
for every H' which satisfies i H'c H,, the equality ks, .. 0
belongs to g; these h' are the. descendents of Hp relative 'to i.
Let I , 2, 3 so that N-- 2 ! . 7.
0, X, 0 belongs to C,.
The open face
O, X,,., = O, Xo. ': 0 belongs to C.
O,
X, O, X, .... 0
does not belong to C,.
9. i4. Canonical parametrization of the set of open. faces. --
E,ery open face is characterized bt a set of independent relations
of the form h ,--H gehere i a H. Con,ersely, to each set
o[ such relations corresponds a [ace vhose equations are all the
Xa, 0 vhere i a H' ½ H and i and H are indices relati,e to one
o[ the gi,en relations.
As an example, we shall now give the set of all the systems of
164
GUSTAVE CHOQUET
relations which define the faces of C.
For brevity of notation
we indicate the relation , c ,_a by writing (il(I. ß H))
It is necessary to add to the systems listed those which follow
from them by permutations of the indices.
Systems including a single relation.
Systems including two relations.
I tt2), (3 2) l(t 2), (2 3, )I I( ! 2, 3), (2[3, 1)f
Systems including three relations.
(il2), (l 3), (21i)I (! 2), (213), (31)13 t .
There is no system of [our relations.
Observe, for example, that the system
1(!2,3), (213, l),
determines the face ),, -,- ), = ),. 0.
9. t5.
Consdquences of theorem 9.2.
Co,o::&aY. I[ an equality of [orm Z z,/(B.)':'O
ho.lds for eve family A,I,,,, of compact subse ts f D, t.here
ernst N constants : such that the linear form ., zs a
linear combination with coefficients of the linear [orms X, as
follows:
HCI JI
In fact, the linear form ZaHXa is 'positive on the cone L,,.
Then it is a linear combination with positive coefficients of the
linear forms which define L..
An equivalent statement is obtained by replacing the capa-
cities f(Ba) by the potentials h(B.) and the linear forms )¾ of
xa by the corresponding linear combinations of 'h(B.).
THEORY OF CAPACITIES i65
Inequalities conCerninõ all operations of the alflebra of sets.
- The inequalities V(B.,_a; A,I ,½") < 0 concern only unions
of the sets A. We have already seen that it is possible to
deduce other inequalities from them which involve intersec-
tions; the following is an exanple:
f(A O B) + f(A f'l B)<f(A) + f(B).
If n compact sets 1AJ are given, we can derive from
then a certain family of sets, in general distinct, by using 'the
operations of union, intersection, and difference of sets. More
precisely, we first form the N½½ atoms as follows:
Then we take
obtain ''1', -- (2
Let 9(f F,,!)
unions Fo of any number of atons. Thus we
--(2 (a'-') ., i) sets.
be an identically true relation whatever
This is equivalent to saying that, if we consider in the space
R'll:, of dimension % the set ½, of the points of coordinates
x, /(F,) when the system of compact sets
arbitrarily, the closure (q of C,, is identical. to
.nutually disjoint.
We have evidently
we show that for
defined by the
as variables not
More precisely, it can be shown that the interior of (..a,, is
identical to the interior of ,,.
The fact that the Fp are not compact sets is not disturbing.
In fact, each of these sets is a K.; it is capacitable, and we can
from now on apply to these sets the inequalities V<0.
The only difficulty arises from the f. act that the E., (which
replace the A) are not arbitrary cap'acitable sets, but are
..,. In order to show that ½q, .'œ,,,
every point (x) ,, there exists a fanily
relations
the A.,
I Ai J varies
the cone
V<0 in which we understand now
but the Ej defined above.
A, 1) may be, where ?(lYpl)designates any continuous func-
tion of the positive variables Y, (p' ,2,..., %).
We assert that this relation is a consequence of the inequa-
liti es V __< 0.
166
GUSTAVE CHOQUET
of mutually disjoint compact sets (E) to which is assigned in
R, a representative point identical with (x).
In the procedure used to compare C, with L,,, we used a
variable base of sets A which were not mutually disjoint.
With the notation thus used we had
But we remark that, instead of considering the compact sets
Ks and their subsets Ka(),.,),. we could just as well have use d
the compact sets with two n&ces Ka. (), with Ka. fqK. p
for i --/= ], and set Ai -. U K..,(,n) with the condition that the
capacities of Ka,(,.) and of U 'K,,,(),a) should be still equal
to ),n.
In fact, it is usually not possible to subdivide'a conpact
set Kn(),a into n compact sets of the same capacity, but this
subdivision can be approximated as we shall now show.
It is sufficient to change as follows the construction of the
Ks. Instead of taking Ksks) a union of cubes, we shall
let Ka(),a -- theboundary.of this union of cubes. Then assu-
ming an __0 given, we shall set
i- t, 2,..., n. In order to show that ½, ..-- ,,, it is essentmlly
this idea that could be used. We shall not show the details of
the proof but give only the result.
i0. i. Tnsoas. Let ?('lxpl ) be a continuous [unction o[
the positi,e ariables (xp (p'- i, 2,..., %). Let us suppose
that [or e,ery index p,'/(Fr) designates the capacity o[ a set Fp
defined in terms o[ compabt sets A (i i, 2, ..., n) by a gi,en
sequence o[ operations U, f'l, ((di#erence .
I[ the relation 9(tf(Fr) t)"__:_" 0 is satisfied by any [amily (A),
the relation ?(lxrl).O is a consequence o[ the % relations
V.__<O in g,,hich each V is considered as a linear [orm of the
,ariables x.
More precisely, ith the notations already' introduced ,,e tiaa, e:
cf, and ½, .... ,.
tl. Possibilities
All the preceding
of extension of the preceding theorems. ,
results apply without modification to
THEORY OF CAPACITIES i67
potentials and capacities
Choquet and Brelot []).
domains which are not
relative to Greenish spaces (see
They apply equally well to plane
Greenish and, more generally, to
Riemann surfaces, taking i the definition of th.e.capacity
the following precautions: f
es of the
study the capact
we
compact sets contained. in a circle of diameter equal to or
less than d, take for kernel Log d/r and for capacity of a
compact set the supremum of the total masses of the admis-
sible measures on this compact set.
More generally, Theorem 7. 6. relative to the successive
differences of the potentials of equilibrium and its corol-
lary relative to the successive differences of the capacities
are 'ex. tended without any difficulty in the proof, to every
capacity associated with a theory of potential in which the
two fundamental principle s 6. 4 and 6. 5 are satisfied.
Such potentials can be defined on a space which is not
necessarily either R o.r even grou. p exemples can be
constructed by replacing the ;D by any locally
omaln
compact space.
Differentiability of capacity. Let D b e a Greenish domain
in a Euclidean space, or more generally, let D be a Greenish
space. Let K be a compact subset of D such that/(K) =/=0
and let ma D K. Let AK be any compact subset of D
contained in the sphere B(m,?) and such that /(AK) 0.
it. t. THEOREM. If ha(K) denotes the ½,alue at m of the
equilibrium potential h(K) o[ K, then
limf(KUAK)-- f(K) ,, [l
'"
am {K)] ' .
Proo[. We shall consider the restriction of the potential
h(KUAK)' h(K) on .K and on AK. This is quasi every-
where 0 on K and quasi everywhere [(l h(K)] on AK, that is,
equal to [1 .. ha(K)] within (where s-- 0 with p). Then
f (K_-X K)--f(K).
,,
is equivalent to the total mass of the measure on (KUAK)
whose potential is I on 5K and 0 quasi everywhere on K.
t68
GUSTAVE CHOQUET
Now this last potential is [h(AK) b(AK, K)] where b(AK, K)
equ.als 0 on AK and h(AK) on K. The potential b(AK, K) is
eqmvalent (for p--,-0)on K to [/(K). b(m, K)], where b(m, K)
denotes the potential of the measure obtained by the sweeping
out process (balayage) on K of the unit.mass at m. The total
mass of this measure on K is a function of m which is harmo-
nic on (D K), and which is 0 on the boundary of D and ! on
the boundary of K and is thus identical with h(K). Thus the
total mass of the measure which generates
is equivalent to f(AK)(1 .-ha(K)); this fact proves the theorem.
tl. 2. Extension of the (}reen'a function. Let P,and P be
two distinct points of D and K,, K two compact sets of
positive capacity contained in B(P,, p) and b(P, ,) respec-
tively. We shall study the behavior of
f(K,) q- f(K, U
when -- 0.
We could use the preceding result, but it is quicker to prove
that this potential is the sum of two potentials U, and U of
measures , and ,.,. each defined on (K, U K.), where the res-
trictions of'U,' on K, and K.: are respectively 0 and h(K,) and
the restrictions of' U on K, and K.are respectively h.(K) and 0.
The restriction of h(K.,) on K can be approximated by
G(P,,P)./(K,). It follows easily that the total mass of ?., is
equivalent to G(P,, P.,.)./(K,).f(K); the same is true of the
total mass of .. Thus
f(K,) + U
2f(K,)f(K)
G(P,, P.) when ? -- 0,
and the convergence is uniform when P, and P..,
disjoint compact sets.
Thus the ratio,
belong to
_.. ,f(K,) + f( K.) --f( K, U
2f(K,)f(Kd
two
defined on the set of pairs of compact sets of non-zero capacity is
a natural extension of the Green's [unction. It is a positi,e and
symmetric [unction of K, and K and can be extended by continu-
ity to the set o[ pairs o[ points of D, and is there identical ,ith
G(P,, P).
EHAPTER III
ALTERNATING AND MONOTONE FUNCTIONS. CAPACITIES.
:12. Successive diffdrences of a function. Let /; be a
commutative semi-group (,0) and , a commutative group.
The operation in g will be denoted by 'r and in by +. Let
ß .
:, ?(x) be a mapping frown into
The successive differences of ?(x) with respect to the para-
meters a,, a.,..., are defined as follows.
x-/,(x; ,)?-() .
(
7"+ x; a,, ..., a,,, a,,+, -
?(X-r .a,), and in general,
)
.... ,X; a, ..., a, ? ..
. (x a,,.,; a,, ...,
In the above definitiøn 'the element x and the elements
are, of course, assumed to be elements of g.
As in the special case treated in
following properties of the function
.diately.
the preceding chapter, the
',, can be verified imme-
12. t. 57,,(x; a,, ..., a,) is a symmetric function of the a;
it is therefore possible to write this function in the concise
form V(x; l(iliEl), or, if Ils a given fixed set, (x; la,).
v(I) v( 'I)
.... .i ai whenever x q- ai '.. x q- a'i
12. 2. x; a -- x;
for each i; moreover, L/(x; l a, l) 0 if, for at least one i,,,
we have (x-rao)'-:-x; (this case occurs when g contains a
.zero element and when a,, is this zero element).
12. 3. If I; contains a zero element 0, we have
w(x; ='"
... v,,.,(0; xl)+v(0;
(0) This means that a mapping og the form c---a T b is defined from $'-' into
with the operation q- assumed commutative and associativeß
i70
12. 4.
GUSTAVE CHOQUET
...,
' .('; -r an; a,, ..., an_ ,,
t3. Alternating functions. , We now make the additional
assumption that g and 5 possess an ordering compatible
with their algebraic structure, and that contains a zero
element. The relations defining the two orderings are deno-
ted by -g and .< respectively.
t3. 1. DEFINITION. A mapping ? [rom g into 5 will be
called alternating of order n, where n is an integer > l, if
\/o(x ; l a] ) < 0 [or each p .<. n and [or e,ery finite family a
,hica is positi½,e, (that is, for which 0 . a [or each i).
The mapping ? ill be called alternating of order co, i[ it
is alternating of order n for each n I.
It is an immediate consequence that if g is an idempotent
semi-group (a-r a -- a for every a), then ? is alternating of
order n if and only if ?(x; l al) 0 for every positive family
lal. I n fact, q(x; a,,..., a,,.)--\/ ,(x; a,,..., a,_,) whe-
never a an_,, since the equation [(X-r a) 'r an
implies the equation [qy_,(X-r an; a,,...,. a_,) -- 0].
i3. 2. Immediate properties. If )is alternating of
order n, then every function qyt,(z; l a (where p < n) is
alternating of order (n .p).
When 8 is such that a-g b implies b 'a.,.c where c -0,
every p(p < n) is an increasing function of , and every
(p n).is a decreasing function of each a. Finally, qYp is an
increasing function of p in the sense that
for J C I and '< n.
The verification of these properties is analogous to that of
the.same properties in the case of the Greenish capacities.
i3..3. Examples of alternating functions.
(i) If g is the positive half of the real axis (i.e.
x > 0) and J is the real axis, then the statement
all points
that the
that ( I)%().< 0 for each n > t.
function y---- ?() is alternating of order. co is equivalent to the
statement that ?() possesses derivatives of all orders and
THEORY OF CAPACITIES
171
(ii) If 1; is 'the class of all compact subsets of a Greenian
domain, and if the operation 'r is the union, then the capacity
f(x) of the element x is alternating of order oc,; the same is
true for the equilibrium potential h(x). (In the latter case, q is
the vector space of all real-valued functions defined on D,
with the classical order structure.)
t4. Set functions.--We shall not continue here the
.geheral study of alternating functions, but shall restrict our
remarks to the case where t is a class of subsets of a set E,
where the operation- r is either union or intersection, and
where is the real axis. It should be remarked, however,
that . some of the definitions and theorems could be easily
extended to the case where is an ordered, .commutative,
topological group.
4. 1. We shall continue to use the term (( alternating for
mappings When -r is union (U); but when 'r is intersection
(f3), we shall use the term (( monotone ) for the function ( ).
More precisely, let g be a class of subsets of a set E and (X)
a mapping from g into the extended set of real numbers (con-
taining q-o and o)("). ; will then be called additi½,e
(multiplicati,e), if from A, g and A} g it follows that
(A, U A) a ; ((A, fq A,) g). Fo r additive g, the differences
with respect to ? will be denoted by V; for multiplicative g
by ] (these .synbols are designed to recall the symbols U
and fq).
A function ? defined on an additive class ; is called alter-
nating o[ order'n if its differences V of orders p .< n are non-
positive (.< 0).
A function defined on a multiplicative class g is called
monotone oœ order n if its differences ] of orders p .< n are non-
negative (? 0).
If we call a function . "defined on I; increasing whenever
(A, A) -- 9(A,) =< 9(A.,.), t follows immediately-from the defi-
nition that every increasing function which is defined on addi-
tive g is alternating of order 1, and conversely- Analogously,
(a) With the understanding that the expressions
[( )---(--ov)] may take arbitrary values.
[(+ >) , (+
i72 GUSTAVE CHOQUET
every increasing function which is defined on nultiplicative
I:; s monotone of order i and conversely.
4. 2. Conjuõate functions.- If ? is a function. defined
on a class g of subsets of E, we shall denote by ?' the function
which is defined on the class g' of the complements X' (E-- X)
of all elements X of 8 by the relation,
+ ?(x) 0.
We have, obviously, (')' ... ?, and (g')' .--g. The two
functions ? and ' are called conjugate [unctions.
It follows inmediately that if ? is alternating of order n
on additive g, then g' is multiplicative and ?' is monotone of
order n on 8'.
t4. 3. Alternatinõ functions of order 2.., If ? is alterna-
ting of order 2 on additive g then is also increasing and
we have,
u ?(. u k) ,, _
whenever aA. and a, A, k,
follows that.
From this inequality it
?(UA,) .... ?(Ua,) ..< Z(?(A,)
whenever ac::Ai for every i.
If g is additive and multiplicative, the two
bel.ow ar.e equivalent.
() q s alternating of order 2.
(if) ? is increasing and satisties.
statements
? (A, U A,) + ? (A, i"1 A,.,) ? (A,) + ? (A._,).
If ? is. a. lternating of order 2 on g, and if ? 0, then ? is also
sub-ad&t,ve, that is ?{A, U a,.,): (A,) q- ?(A.,). We shall not
prove these elementary properties which have'in large meast/re
been proved in the preceding chapter.
14. 4. Monotone functions of order 2. If ? is mo.notone .of
order 2 on multiplicative 8 then from the' properues of .ts
conjugate, ?', the corresponding properties for ? can be deduced.
We find that ? is increasing, and
?(A f'l k)
?(-).
THEORY OF C.kPACITIES
173
whenever aA, and
whenever a A for all i.
When g is both additive and multiplicative, the following
two statements are equivalent:
(i) ? is monotone of order 2.
(ii) ? is increasing and satisfies
?(A, U A,) + ?(A, fq A, 9(A,) + ?(A,).
If ? is .monotone of order 2, and if ?() -- 0, then 9 is supra-
additive in the sense that ?(A, O A) .9(A,) + ?(A) whenever
(A, f3 A,) .
14. 5. Alternating and monotone functions of order 2.
Ta.oaz. If g is both additi½,e and multiplicati½,e, then
e½,ery [unction (X) defined øn 8, ,hzch is both. alternating
and monotone o order 2, is increasing and satisfies
?(A, U A,) + ?(A, fq A,)--. ?(A,) + ?(n,).
Coneersely, if a function ? defined on c, zs ncreasng and satisfies
the abo,e relation, then, [or e½,ery n___-' t,
V(X; taxi) -- ?(X fq a) ?(a) <0, where. a -- f'] A,
A.(X;IA, ) where A--JA,.
The [unction q, vhich is thus seen to be alternating and mono-
tone of all orders, is called additive.
Proof. If .9 is.both alternating and monotone of Order 2,
then we obtain simultaneously,
9 (A, U A,) + ? (A, f A:) ..< and
whence the equality of the two nembers.
> ?(A,) + ?(A,),
Let us assume now
that is increasing and that .the .above mentioned equality
holds 9 Clearly, this equality mphes
(X)--?(X U A,) .-- ?(X fl A,)--- ?(a,).
and
hence V, (x; A,) ? (X fl a) __ ? (a), where a A,.
174 UST.V. CHOQUET
We now assume that the relation. Vp :(xna! holds
for all orders p _< n and we Prove t for p--n- .
If a A, and a' .: 0 A,, then
A,, ...,
n a),,,
n a)... n u
V,,.,(x;
From the fundamental equality, the last expression is equal
to ?(Xna'). 9(a'), which is indeed the desired quantity; it
is obviously non-positive.
The second relation for the A is deduced by duality from
that for the V,.
5. Capacities. Let E be a topological space, g a class of
subsets of E, and ? a mapping from g into the extended real
line [.
t5. t. Continuity on the right. We shall say that ? is
continuous on the right at A (A a g), i[ [or e,ery neighborhood
W of ?(A) there exists a neighborhood Vo[ A in E, such. that'
whenever
X g and
AcXcV
Obviously this definition may be applied also to the case where
?(A), .-,, + oo or ?(A) --
If; !s cont!nuous on the r!ght at every A a ,';, we shall say
that s continuous on the right on g.
onlg5. 2. flapacit on a class g., of sets. A [unction ? defined
is called a capacity On 8 ? is increasing and continuous
on the right on g.
We shall now define the following functions of subsets A of E.
Interior capacity o[ A "' ?,(A)--sap ?(X) (for X a g and
X A). When there exists no element of g contained in A, we
set ?,(A) -- inf ?(X) (for all X a g).
In particular, ?,(to) is thereby defined for every open set o,
and we can now define for any A:
Exterior capacity o[ A -- ?*(a)--inf ?,(to). (to ope. n and a .o).
We have always, ?,_<?* and ?,, ?* are ncreasng funcuons.
A set A is called capacitable if ?,(A) ?*(A). It is a trivial
TIIEORY OF CAPACITIES
conclusion that every open set is capacitable. We shall
consider only capacities for which every element A of .g is
capacitable. ' This will occur in particular when g is absorbing:
A class g of subsets of E is called absorbing if for every open
subset o0 of E and for every pair (A,, A) of elements. of g such
that Ai C:to (i-- t, 2), there exists an element A of 8, satisfying
(A, U A).,A.o. o. For instance, g is absorbing when it is
additive.
For simplicity, let us assume that ?(A)is finite. For
every , > 0 there exists, by virtue of the continuity on the right
of ?, an open set o such that A o, and 0 (A') .-?(A) for
every A' satisfying A c-A'
Moreover, since g is absorbing, to every B a ,; and contained
in to, there corresponds a C a g such that (A U B) cC co. Hence
?(B) =< ?(C) < ?(A) q-
and therefore, ?*(A)?(A). Clearly, since .moreover?*(A)--?{A),
the element A of t.'; is capacitable. There s therefore no contra-.
diction, when for arbitrary capacitable sets A we define
?(A) -- %(A) ,-- ?*(A).
_t5. 3. Alternating capacities. We shall introduce a scale
of classes of capacities.
A capacity ? on 8 is called alternating o[ order cx. i[ t; is additi,e,
(a restriction which is not essential [or &,,,) and i[ ? satisfies
one o[ the [ollo,ing conditions a:
ex,.: I[ I A,,I is any increasing sequence of subsets of E,
then ?*(a,,)--q*(A), where A [.JA,.
c,,: Gi,en > O, there exists an > 0 such that the inequa-
lity
?(A,) ..... 9(a,)-< 'q
the inequality
?(A, U A.)
(ai cA, ai and Aa t.'; with i t, 2) implies
The [unction q> is alternating of order n (n-: 2, 3,...,).
:The [unction is alternating of order c.
t5. 4. l1onotone capacities. -- i capacity ? defined on g.is
called nonotone o[ order qlr i[ is multiplicati,e, (a restinc'
tion ,vhich is not essential for ql[,,,), and if satisfies one of
the following conditions ql[:
176
GUSTAVE CHOQUET
xo,,,: .If I is any decreasing sequence of subsets
then .,aee -- fqA,,.
,;b,, : Given > O, there' exists an '½ > 0 such that the
litg
of E,
ne q ua -
?(A,) --?(a,) .. 't i (a, A,, a, and A, g, s, ith i 1, 2)
implies
?(A, f'l A..,)- (a, f"l a) < .
Ab,, The [unction 9 !s monotone of order n (n
,ilb©:: The function ? s monotone o[ order
--2 3 .)
-15. 5.
Immediate consequences of the
.,)b,, +, =.;lb,, for
,, +, ct,, and
definitions.
n2
The above relations are an imnediate consequence of the
properties of functions which are alternating or monotone of
order n> 2 (stud!ed at the beginning of this cha.pter). F. or
example-he relation .I.,I.,..,=.lil,,, derives from the nequahty
An important theorem which will be proved in the sequel,
states that in very general cases the following relations hold:
and
t5. 6. Conjugate capacity of a capacity. If ? is a capacity.
defined on a class g, Which is assumed to be absorbing, then 'the
conjugate function ?' corresponding to ? is not in general a
capacity because, firstly, l;' is not in general absorbing, and
secondly, ?' is not in 'general continuous on the 'right.
However, if g is an absorbing class o[ closed subsets o[E
then for every capacity defined on ; another capacity..
may be associated with it which is also defined on t;. This s
done by setting '9(X)--. !l i. ) for every X a g. The d. efi-
nition is neaningful since X an open set and hence a set
for which ? is defined.
The function is obvious increasing. It is also continuous
on the right. This is due to the fact that by the definition
TtlEORY OF CAPACITIES .77
of 9,, every open set X contains closed sets elongmg to &;,
such that their capacity differs frown that of I X by an arbitra-
rily snall value. Hence (X) is a capacity.
Let us further assume now g is the class of all closed subsets
of E (E is additive, and. therefore absorbing). For every open
subset to of E we have,
Clearly, inf ?(G) -- ? ( I to) 9'(o),where 9' is the conjugate
function corresponding to 9, ? "being defined on the class g' of
all open subsets of E. It follows'that }(o)-
'As a definition, the function } will therefore be called the
conjugate capacity of 9'
Moreover' for every X g we have
It can also be immediately verified that for every AE,
?,(A) + "*( I A) -- 0 and ?*.(A) + }'( l A)..-- 0.
Thus the operation b (conplementation), establishes a
canonical correspondence etween the ?-capactable and the
,-capacitable sets.
15. 7. If. ?.is of order b,, (a,.), then -'9 is of order
(b,,). Ths s an immediate consequence of the last two
equalities.
I[ ?isof order,lb, (for ,---(t,b) or a-- n> .2), then.} is of order
a,. For the proof of this correspondence it s sufficient to show
that the fundamental inequalities which define a class al, still
hold when the closed sets are replaced by open sets, a result
obtained without difficulty from the following lemma.
15. 8. LzA.- - Let Io, l iEi ba a finite family of open subsets
of E such'that ? (o) is finite for.each i. To each > 0 there corres-
ponds a ' [amily I XI½ o[ closed sets, ,,ith X o [or e,ery i,
and such that ?(nr,) .... (nx,t<,'or ei, ery JI.
ß fiEJ iEJ '
78
GUSTAVE CHOQUET
In fact, for arbitrary J cz I, consider a closed set
such that 9(o)
If we set X
Xj (oj --- N
J
- <
U Xj, we obtain Xi c½oi.
On the other hand,
iJ
N XD Xj and hence the sets X satisfy the desired relation.
i5. 9. If ? !s of order Ao(. l,b o.r n '2), the,n y
is not necessarily of order qlt,, except n the case whe E s
a normal space.
In this case it can be shown (see next Chapter i7..9. and
17.10.) that the inequalities defining a class A0 are still valid
if the closed sets are replaced by open sets. The inequalities
which define the class lb are then obtained by complemen-
tation. Thus we see that a perfect duality does not exist bet-
ween the alternating and monotone capacities. This is due to
the fact that the definitions of ?. and ?* are not parallel; can
?'
be defined only after 9, has been defined.
CHAPTER IV
EXTENSION AND !ESTiIC:TION OF A CAPACITY
i6. Extension of a capacity. Let $ and $ be two classes
of subsets of a topological space E such that $ $, and let f
be a capacity on $,. We shall always suppose $ to be such
that each element of g, i.s f,-capacitable, which is the case, as
We have seen, when g, s absorbing (for example, additive).
16. t. DrrNITIOrq. ,,,, The [unction [. on g defined by
/a(X) f(X) [or each X a g is called the extension of f, to g.
It is indeed an extension in the ordinary sense for if X
e h%e f(X) '- (X) -- f,(X).
This function f, is a capacity. First, it is obviously increa-
sing. On the other hand, for each A E such that, for example,
[, is finite, and for each > 0, there exists an open set
containing A and such that f,() F, (A) < a; hence, if
A a g, we have the inequality f(B) ../(A) < a for each
B g, such that A B o. This fact shows that f is conti-
nuous on the right.
Since g, g,, we have/,.(x) for each set X. But
:this inequality obviously becomes an equality for open sets.
It follows that (X)- f(X) for each X. In particular we
have, for A a g, [7(A) -- [(A) ' (A), and it follows that
A is/-capacitable, althougk we have made on g no restrictive
hypothesis such as g is absorbing .
It also follows from these relations that if an X E is f,-capa-
citable, it is also /-capacitable and we have/,(X)-/(X).
In short:
i6. 2. THEOREM.
Capacity and
,,, The extension of a capaeit f, is a
ft-f;.
There are more [,-capacitable sets than f-capacitable sets.
80 GUSTAVE CHOQUET
An example of extension. , ..... If we take for 8 the class of
all the subsets of E, each X E becomes f-capacitable. This
example shows clearly that it is not of interest to make exten-
sions to .classes which are too large. Of course, extensions
enrich the class of capscitable sets, but we lose preciseness in
the process since now f, [.
16. 3. THEOREM. If X E is such that each element of
co. ntained in X is f,-cap9citable , or is contained in .an [,-capa-
ctable subset o[ X, e h'a,e /,,(X) :... /,(X); thus, i[ this X is
[,.-capacitable, it is also [,-capacitable.
In fact, if A B X with A a g and f,,(B)- f,*(B), .we have
f(A)-- f, (A) .. ..__
f ,(B) --,. [,,(B) :[,,(X). By companng the
extremes it follows that f,,(X)f,,(X).' Since we have
already the inequality [,,_<f,, we-av e, indeed, the equality
f,,(x) L,(X).
t6. 4. Applications of theorem i6. 3.
6. 5. First application. Suppose that each element of
g, is/,-capacitable. The preceding theorem is applicable then
to each X c: E. Therefore we have the identities
and ff. In particular, the f,-capacitability is identical
to the /ccapacitability.
ExxMrnS t. If is the class of f-capacitable subsets
of E, it is 'the largest extension of f., .which does not change the
capaCitability. We shah say that t s the canonical extension
of f,.
ExxeL. 2.. Suppose that there exists a closed set N E
which contains each element of g, and tha)t, for each element
A g,, the set Af'IN is f,-capacitable. '
Then each element A a g is /,-capscitable.
In fact, we have/,,(X) --/,,(X f'l N) fo r each X E. Fur-
th.ermore, for' each open set o, we have f,( U I N);
ths shows that/:(X)-/:(Xf'IN) for each/ .. o
Thus each X such that
/,-capscitable.
We have, for example, one
XN is /,-capscitable is also
such circumstance in taking for
g, the set of closed subsets of a closed N of E and for g the
set of all Closed subsets of E.
THEORY OF CAPACITIES
i8i
16. 6. Second application. Suppose that each element ofg
contained in an element of g, is an element of
EXAMPLE. Let $ and $, be hereditary classes of closed
subsets of E. Then theorem i6. 3. is applicable to each X
which is a subset of an element of g, that is, f,. .. f. for
this X.
Special case. If there exists a subset N E which
contains each elexnent of g, and if each element of ga contained
in N is an element of g,, the theorem is applicable tO each
X N. We obtain an example of this situation by taking g,
as the set of all compacts contained in a set N in a Hausdorff
space E and g as the set of all compacts in the same space E.
16. 7. Transitivity of the extensions. If 8, g, g., if f, is
a capacity on g, and f, f its extensions to ga and g, respec-
tively, it is obvious that [. is identical to the extension of [, to
g. Indeed, the exterior capacity of a set remains invariant
in each extension. This amounts to saying that the extension
is a transitive operation.
t7. Invariance of the classes c by extension.
i7. i. Classes a,, ,. .. From the identity f7- -f it follows
immediately that, if f is in the class a,., then' each extension
of f, is in the same class.
t7.2. Classes of order a lreater than (i, a).
to study these classes, we will need a new definition.
In order
t7.3. DEFINITION. ß A class g of subsets of E is' rich if, for
each. couple of open sets w, o of E and each element A of ,g such
that A (o, U o), there exist two elements A, and A of g such
that A, to,, A o, A A, U A..,.
17.4. EXAMPLe... If E is a normal space, each. her?di-
.tar.y set. of. closed subsets of E is rich. In fact, by duality,
t s sufllcent to prove the following: if F, and F are two
closed sets of E, and if G is an open set such that G:D F, Iq F,
182 Gusrxv. CHOQVET
then there exist two open sets G and G which contain G and
such that
G :D F
G,. D F, and G- G, f G.
Now the sets (F, G) and (F. G) are closed and disjoint;
the normality of E implies that they are respectively con-
tained in two disjoint open sets g, and g. It is then sufficient
to take G, G U g, and G . G U g,; these open sets have
the desired' properties.
17.o. EXAMPLE ,, More generally, if is a hereditary
class closed sets'of E such that each element of g is normal,
then. g is rich. This example is a generalization of the pre-
ceding one. The preceding proof.is applicable provided that
the passage to the complements s made xvith respect to the
closed set A of g which is to be covered by A,. and
17.6. EXArLr... .... If g is a class of compacts of E such that,
for each K a g, each compact k K and each neighborhood
V of k, there exists an element X of g such that k X V,
then' g is rich.
In fact, when the open sets to, toa are given as well as the
omp.act K . g such .that K ( tJ oh), we find immediately,
y using the nformatlon in Example i7.5., two subcompacts k,
and k of K such that k o, k. cz o, and K k, O k. By
virtue of the hypothesis, there exist K, a g and K, a :; s'uct
that k, K, o, and ka Ka oa. These compacts form the
desired covering.
t7. 7ndLTe7,. -- Let f bea capacity on a clas. s g of subsets
of E, i X, l,a be a finite family of arbitrary su. bsets
of E,,ith f*(, X,) finite for each J I.
For each > 0, there exists a family I tol of open sets of E
such that Xi' to for each i and
for each
In fact, for each J I, 'there is an. open % such that
arid
THEORY OF CAPACITIES
i83
If we set oi= N oj, the family 10oili obviously has the
desired properties.
i7. 8. LEMMA. Let f be a capacity on an additi,e and rich
class g o[ subsets o[ a space E, and let '! øI , be a finite Jamfly
ofopensetsofE, ,ith[(o
there exists a Jamfly
)finite [or each J I. For each
A,J,a, of elements of g such that
A,.. ½ for each i I, and such that for each J I, ,e ha,e
(Note that the restriction that the f(½o) are finite is not
tiM; we would have an analogous statement if some
f(ai) were-- or q-).
For each J c I, let A be an element of g such that
essen-
of the
Aj (22 U
and such that
By using the fact that g is rich we can, for'each J, cover
A by a family of elements A. ia, of g such that A.
for all i J. The proof follows immediately if J contains
only two indices; in the general case we apply the same process
repeatedly (exactly (--i)tmesy. Then for each i-I let
It follows immediately that the family i A has the desired
properties.
17. 9. LEMMA. '' Let f be a capacity on an additive and rich
class g of subsets o[ a space E. Let I be a finite set of indices
and q). ( i xJ ) a continuous real'Junction o[ real variables xj(J c I).
If for each family A, l,e, of elements of g e ha,e 4)( lxl ) 0
where :r.,-f(, A,), e hae the same inequality g,,hen e
replace the sets A by arbitrary subsets of E and each
xj by
13
18 USAV
In order to simplify the proof we shall assume again that the
capacities which occur are all finite. In order to include the
c.ase where they are not, it would be necessary to give a pre.:-
cse definition of the continuity of (I) at infinity. This deftre-
tion is easy to formulate in the particular case (case of q' linear)
where we shall have to use it.
The inequality (I)( l.SlJi ).,0 is satisfied ,vhen we take ele-
ments of 8 for the A, therefore also, by virtue of lemma .17. 7.
and the continuity of q), when the sets A are open. Lemma
t7 9 then follows because of the continuity of , from. Lemma
t71 . which asserts the possibility of approximating in a
suitable way each of the A of a given family by an open set
17. 10. APPLICATION. Let E be a Hausdorff space
containing a countable sub-set which is everywhere dense,
and let œ be a positive, sub-additive capacity defined on. the
class g .q;(E) of all compact sub-sets of E such that/(X) 0
whenever X contains not more than one point.
Then there exists a sub-set A' E which is a G e,eryVhere
dense in E (henCe A is a residual of E mhen E is a complete
metric space) and such that f,(A) if(A) = 0.
I ' be a countable sub-set
For, let D . a,, a,...., a,,, ...
which is everywhere dense in E, and s an arbitrary positive
number.
There exists, for each n, an open. set o,such that f(m,,)
and a, o.. .
If we set fl -- [.J o and œ'1 [.j o,, then, from the abo,
11 . .....- r
On the other. hand, since the sequence , is increasing, and
since each element of g is compact, it can be easily shown that
f(P.) -- lim f(,) (see end of 28. 2., Chap IV).
It follows that f{)<: s. Now, 9. is an.open set which is
everywhere dense in .. HenCe, there exists a sequence of
open sets G,, which are everywhere dense in E, and whose
capacities tend to 0. Their intersection is the desired set A ('-).
(-2) Mazurkiewicz [1] has proved a weaker result, concerning only the interior
capacity of A, whenever E is a compact sub-set of a Euclidean space, and f is the
Newtonian' capacity.
THEORY OF CAPACITIES
17. . Tasoas. If f is a capacity of arbitram d ordera on
an additive and rich class g o[ subsets o[ a space E, each extension
of f to an additive [amily is also of order
Proof.. For the class a,.,, we have already seen that this
statement is satisfied., even without assuming that g is additive
and rich.
For ?> (l, b) it is sufficient to remark that each class et is
defined by a system of inequalities of the form
is a continuous function of capacities .A These ine-
qualities remain valid, according to Lemma 17.9., for the
exterior capacities f* (y Ai, where the A are (før example)
elements of the set g, on which the extension [ of [ is defined.
Since L( U A,i & by the definition of f,.,, thd ine-
\, iEJ / . iEJ /
qualities :> 0 remain true for f,,.
17. 12. COROLLARY. ' If a capacity f on an additi.e and rich
class g is of order cx.(n > 2), each of the inequalities V. 0
(p n) can be extended to the exterior capacities o[ arbitrary
subsets of E.
This corollary.
Lemma t7. 9.
actually an immediate
consequence of
18.
lnvariance .of the classes
1 by extensio n.
18. . The class ;b,,,. If a capacity f, is of order ob,,, on $,
we have f,,(A:)--[,, ( A,) for each sequence a,,. However,
since we know only that [,[,,, we cannot show that
f.,,(A,)--/,,(ff]A,). Therefore, the order.'t,,,,,, is not conserved
by extension.
18. 2.
Classes ,,tb,. for _: (l, b).
18.3. Ls,. Let f be a capacity, on an additive and mul-
tiplicative class 8 of subsets of E, and let Xf, be a finite
(a) This statement is less obvious for the class , v. However, notice that the
condition which defines C%,b may be formulated as follows: for a.cA and
,,.c .½., o h,,, I(,, u ,.,) . l(,,, u ,,.)...<.. 'rj.(/(&) . tim)), (I(A,) - l(,,,))'! where
tF{u, v)--0 with u and v.
i86 GUSTAVE CHOQUET
family of subsets of E such that f, X, is finite for each
J I. For each O, there exists a family ! A, l,.of elements
of $ such that AiC X for each i, and such that, for each J c I,
e ha,e
Indeed, for each J I let Aj be an element of g such that
and f X, -f(&),<..
for each g a J. This family obviously
stated.
Then let A,: U As
satisfies the condition
t8. 4:. L.MMA.
tiplicative class g.
i7. 9., if ,;,e ha,e' (D( lxsl).'.> O, with x.,- f(,O *")' for
. Let f be a capacity on an additive and mul-
With 'the same con½,entions as in Lemma
each
choice of the family I A, I of elements of g, aae the same
inequality hen e replace the A by arbitrary subsets X of E
, ii!J ,.
This lemma is an immediate consequence of the continuity
of q and of Lemma 18. 3.
i8. 5. DEFINITION. ' A class ; of subsets. of a topological
a.p.a.ce. E is called G-separable i[ [or each couple X, and X, o[
easlont subsets o[ E each .of "'hich is either an element of or the
intersection of one such element ith a closed set o[ E, there exist
t,o disjoint open sets to, and to of E such that X, to, and X
The following are examples of G-separable sets
18. 6.
Any class 5 of compacts in a Hausdorff space E.
18. 7. Any class of closed sets in a normal space E.
It is obvious that, if X}½I is a finite family of mutually
disjoint sets each of which is either an element of or the
intersection of such an element with a closed set of E, then
there exists a family i oil of open sets of E such that
XC2 o for each i and ofqo- ' for i ,: 1.
THEORY OF CAPACITIES
187
18. 8. L,. Let f be a capacity on a set of subse of E,
and let X,},a be a finite famil of subse of E such that
each if(X,)is finite, where X, X, (J I).
iJ ß
When all the Xs make up a G-separable. set , there exists for
each a 0 a family of open sets l to, I, of E such that X, o, for
each i and
f(o) f*(X)..< for each J I, where
oj --- N
Pl'oof .
such that
for each
We can easily construct a family of open sets .
f(.,) f*(X,).
whenever J cz J.
This family plays a transitory role in the construction.
First let o -- . Then we sup. pose the o-defined for all J
of cardinal number 'J > p, and n such a way that for each
such J we have
J'2)J
For each J such that J '-- p, we then define
These Yj thus defined are mutually disjoint; they are therefore
separable by some open sets G which one can in addition
restrict by the condition G P,. We then define o as follows:
It is obvious that the family of o thus increased
possess the three properties stated in (1) above. We continue
the construction until we obtain the o with J- ; they are
the desired o.
.9. COltOLLAR, Y, OF LEMMAS 18. 4. AND i8. 8.
capacity o n an additive and multiplicative class g.
Let f be a
With the
GUSTAVE CHOQUET
same com, entions as in Lemma .18. 4, we ha,e the inequality
4p ( I a:j I ) > O, where xa .--f*(Xa) and Xj X, for each family
;EJ
X} , such that the set of X., is G-separable.
This corollary is an immediate consequence of lemmas 18. 4.
and 8. 8. and of the continuity of (I (we use Lemma t8. 4.
in the particular case where the X are open sets).
18. 10. THEOREm. If f is a capacity of order ,[!o(c/.._ l b)
on an additi,e and multiplicati,e set g,, the extension of f to a
multiplicati,e set g,. is als ø o[ order !o, ,hen the set t;,. is G-sepa-
ruble ([or example, if each element of ga is compact and E is a
Hausdorl space, or if each element of ga is closed and E is
normal).
This theorem is an immediate consequence of corollary 18. 9.
i9.
Extension of a class g by a limit procedure.
We are now going to study the extension of a .capacity [ in a
case where the set g,_ is deduced from g, by a process inde-
pendent of the given capacity f.
19. t. THEOREM. J Let , be a multiplicati,e class o[
-compacts o[ a space E, and let g. be the set o[ arbitrary intersections
o[ elements of g,. if f,. is the extension to .g, of an arbitrary
capacity f, on g,, then for each A
L (A,.,) inf f, (X)
(A X; X a.
If f, is of order (.'2> , b), then f. is of the same order.
If is additi,e as ell as multiplicati,e, h the same pro-
perty; if then is of order , f is of the same order.
Proof. We use the fact that., for each A a g and for each
open set ½o containing X, there exists an element B a g such
that A B ½o. This statement is an immediate consequence
of the fact that A is the intersection of a filtering decreasing
family of compacts which are elements of g.
It follows that for any finite family Ai½ of elements of ,
and for any two families of open sets , and j such that
Ai o., U Aifla for each JI,
THEORY OF CAPACITIES
i89
there exists a family
indices i and J
of elements of g, such that, for any
These relations show that we shall be able to approximate
each finite family of elements of &. from above by elements
of g, in such a way that this approximation is preserved by the
operations of intersection and union. The formula
fo. (A) -- inf .f, (X)
X; X
follows from the fact that fo. is continuous on the right, that
[, (X) -- f (X), and that we can approximate A. from above by
some X.
'Henceforth, for each inequality
where
which is valid for f,, it is sufficient to carry out a passage to
the limit in order to obtain the same inequality for [. This
remark establishes the second assertion of the theorem.
When g, is additive, the additivity of g follows immedia-
tely, and the process which we have just used for f(gA)
is also valid for/(igAi)..This fact proves the last part of
the theorem. '
20. Restriction of a capacity. Let g, and g be 'two classes
of subsets of a space E with g, g, and let f be a capacity
on g.
The restriction o[ [., to g, is the [unction f, defined on g, by
the relation. f, (A) [ (A) for each A
It follows immediately that f, is a capacity. We suppose,
as everywhere else, that the given data are such that for
(and f) every element of g, (respec. t!vely )is ca.pacitable, or
example, because and are addhve or absorbing.
The following relations hold for each X E:
f,,(X) h, (X)
(X) f; (X).
If f is of order cx,,,(.$b,.,), we cannot therefore conclude that
[, is of the same order. But, if g, and g are additive (multi-
.90 GUSTAVE CHOQUET
plicative), and if L is of order t (with ? (i, b) (or respec-
tively of order ,), it follows immediately that is of the
same order.
This operation is interesting in a special case.
20. t. Special case.
each open set o c E, the
if N be open).
If we take [or g, t of elements of
[or each X N the equalities
Let N be a subset of E such that, [or
set (Nfo)is [-capacitable ([or example,
included in N, ,e ha,e
f,, (x) f,. (x) f,* (x): (x).
In particular, each subset X of N is simultaneously [, and
f,-capacitable or non-capacitable.
The first of these relations follows immediately. In order to
show the second, we shall suppose, for example, that (X)
is finite. For each s > 0 there exists an open set o such
that X co and f, f(x) < Now we have the follo-
wing sequence of inequalities:
(X) < (X)< (N f'l co) . [,, (N f'l co) -- œ,, (N Flco)[,, (co).,-- f, ((o).
It follows that (X). a for each a > 0, and the
desired result follows.
20. 2. Application of the preceding operations. We shall
use these operations especially in the study of the capacita-
bility of sets. In fact, it is often convenient in this study
to suppose that the space E and set g possess a certain regu-
larity. The operation of restriction will permit us to
replace E by a subspace N; then the extension operation will
permit enrichment of the new class g, thus obtained, a step
which often proves useful.
20. 3. ExArns. Let E be a Hausdorff space, and let g
be an additi,e 'and hereditary class of compacts of E. Let [ be
a capacity on g.
Let X be a subset of E such that every compact contained
in X is/-capacitable; and suppose that there exists a completely
regular set N such that X N E, and such that each
subset of N which is open relative to N is/-capacitable (if X
TllEOR'¾ OF CAPACITIES
ß possesses a completely regular neighborhood, we shall take
for N the interior of this neighborhood;if th.e elelnent A. of g !s
netrizable, we shall take for N the set A ff the capacity [ is
such that each K, of E is capacitable).
We ;,ish 'to sho, ho e can replace the study of interior and
exterior capacities o[ X by the same study in a simpler case.
Let 8 be the additive and hereditary set of elements of g
contained in N. If [, is the restriction of [ to g,, w.e have
/,,(X') --/,(X') and ff,(X') f*(X') for every X'c N.
Observe, on the other hand, that we can consider [, as a capacity
on the set g, of subsets of the space N; we obtain for each X' N
the same values for the interior and exterior ,-capacities when
we consider 8, as a class of subsets of E or of an arbitrary
space in which N is imbedded. This reinark will allow us tO
imbed N in a new normal space as follows:
Since N is conpletely regular, it can be imbedded in a compact
space F. Designate by t;. the set of its compacts and by [. the
extension of f, to
According to theorems t6. 2. and t6. 3, as every compact
included in X is f-capacitable, and the n also f-capacitable,
we have,
f,,(x) _ f.,,(X) f,(X) = f,(X).
It follows that the interior and
exterior.capacities of X are the
rem t7. tl. sho;vs that if f, is o.f order a., f, is also. And
Theorem t8. t0. shows that if f, is of order }/,. with a> (1, b),
'then f is of the same order.
same fo}f and for/,.
No½,, has the a&,antage o[ being a capacity defined on the
.set o[ subcompacts o[ a compact space.
Let us shog,, in addition that i[ [ is of any order a. or of order
.,b. ,,ith g > (i, b), the capacity f, is of the same order.
It is obvious that [, is of the same order as [. Then Theo-
CHAPTER V
OPERATIONS ON ;APAC;ITIES AND EXAMPLES OF CAPACITIES
In this chapter we shall study first some operations which
transform capacities of a given class into capacities of the sane
class, and then several examples of capacities, some of which
are important and will be used in the following chapters.
2[. Operations on the range of capacities.
21.1. If (1)(l,l) is a continuous, increasing function of the
real variables w (i a I), and if (f)at denotes a finite fanfily of
capacities defined on a class t; of subsets of a space E, then
the function /(X), defined by /(X)-:-(l/(X)i)'is a capa-
city on g, and we have
If each of the f is of
and
order a,. ,(lib,,,) then the same holds for/k
If (I) is a linear form with non-negative coefficients, and if
each [' is of arbitrary order eL(.,tl,.), then the same is true for [.
2. 2. If (/a) is a sequence of capacities defined on the same :;,
and if the [, con,erge uniformly on f?, to a function [, then this
function is a capacity. The . converge uniformly to [,, and
the fl converge uniformly to If' each f. is 'of order
then [ is of the same order.
2. 3. If ([,) is a decreasing sequence of capacities defined
on the same ,g, then the limit [ of this sequence is a capacity.
We have /,.,limf.,, but not necessarily: f, .'-lim f,,.
If each f, is of class et(.l,), with > (, b), then the same
holds for the limit f.
THEORY OF CAPACITIES
t93
We shall not give the very easy proofs of these statements.
2. 4. If (1)(u) is' an increasing conca½,e [unction o[ the real
,ariable u, and. if f is a capacity of order a,,. on an additi,e class
then the function g (I)(f) is also a capacity Of order
Proof. The assmnptions on (D imply its continuity; hence g
is a capacity. Let us sho;v that V.(X; A, B)_<O. We know
that t.., (X; A, B)y<O and that the ',-7, and qfa with respect
to the function (1 are non-positive 'since q) is increasing and
concave. If we set
V,_,(x; A, B)y Xau, V,(x o A; B)f
V,(XUB;A)g....Xa, .. f(X):
then the X, ),,,, ),., are non-negative and we have
f(x)-
f(XUA)- k,, + X +
f(XUB)- k,, + X,, + X,
.f(XUAUB) Xo q- X.. + ),,, q-
kAB.
If we add the two relations
and
term by 'term, we obtain
x..,+ x.,,,)-,v(z0+ x.,.+ x,+ 0,
which may be written also as follows:
V.,.(X; A, B):,<_O.
2t. 5. Generalization. An analogous result is obtained if (P
is replaced by a function of several real variables whose V?,
and 7 are non-positive.
More generally, one could shog½ that by composing t,o alter-
nating [unctions o[ order n (in 'the sense o[ Chapter ), the
resulting [unction is alternating o[ the same order. The proo[
o[ this last result is not simple; gee shall not give it here.
21.6. If )(u) is an increasing, com, ex [unction of the real
oargable u, and i[ [ is a capacity o[ order 4 on a multiplicati,e
class g, then the function g -- (I)(f) is also a capacity o[ order
194 aVSTXVE CaO9VT
This statement is equivalent to the preceding one, for observe
that, if we set '(u)- .(. u) and [' ...... [, then the
function ' is increasing and concave, and [' is alternating of
order 2 on the multiplicative semi-group g.
An extention analogous to that of 21.4, can be obtained in this
ease also.
7. If f a capacity of order a.(n 3) on an additiee class
t;, then V,(x, A,) is a capacity o[ order a,_, on g for every
A, ag.
The continuity on the right of V,(X; A,)is obvious. On
the other hand, every difference of order (n . i) of this diffe-
fence V, is a difference V, of f; it. is therefore non-positive.
An analogous statement eoneermng the capacities of order
½'?
.lb,(n.3) on a multiplicative class , is obtained if V is
replaced
Change o! variable_ in a capacity.
22. 1. .Let E and F be two topological spaces and ; and
t,o classes o[ subsets o[ E and F respectively. A mapping
Y -:.:., ? (X) [rom g into vill be called increasing and continuous
on the right i[.
a) (A, c- A..,) ((A.,) ?(A")'), for any elements A,, and A
of
b) [or e,ery neighborhood V, oF ?(A,), there exists a neigh-
borhood U, o[ A, such that the relation ?(X)c--V, holds for e{,ery
X a such that A., U,.
I[ [ is a capacity on S, then [unction e(X) --/(Y), vhere
Y = ½(X)g,,ith X a g, is ob,iouslt a capacitor on g. We shall
say titat e is derived frown/by the change of variable Y '-= ½(X).
22. 2. .EXAMPLE. "' Let y" ?(x) be a continuous mapping
from E into F For any class ':
ß ,., of subsets of E, we shall still
denote the extension of 9 to g by 9, and let be the image of
½?
,, by ?. This mapping ? from f, onto , is increasing and
continuous on the right.
THEORY OF CAPACITIES
i95
If f is a capacity on 1, then for every B F we have
e,(?-'(B)) f,(B) and e*(-'(B)) <f*(B);
and the last relation is an equality if the mapping-? from E
into F is an open mapping, that is, if it maps open sets on open
sets.
More generally, the following relations hold for every A E:
e.(A) </. (?(A)).
and
e*(A) f*(9(A)).
.An important special case is the following:
For E we take the product space of two Hausdorff spaces F
and G; and let ? be the canonical projection from E on F. Let
Us suppose that every element of g is compact, that = (g),
and that the condition ?(K) a i.nplies K a g f.or every comp. act
subset K of E. Using the notahon employed n the prece&ng,
we assert that the relation e(o) holds for every open
subset o of E.
Indeed, for every compact subset B c ?(to), . there exists .a
compact subset A c o such that B = ?(A); ths statexnent s
easily deduced from the fact that B is compact. If we take,
for the sets B, elements of i whose -capacity approaches that
of ? (o), we obtain e(½o) __/(?(to)); but we know already that
e(to) <[(?(o)), hence the equality.
It follows that e*(X)-/*((X)) for every subset XE.
Since we know already that e,(X) f,((X)), the e-capacita-
bility of X, that is, the condition e,(X)= e*(X), implies
f*(9(x)) _<_ f,(9(x)), whence f*(9(x)) ,-
22. 3. TuUOUUM. The e-capacitability of x implies the
[-capacitability of its projection ? (X), and,$,e a,,e e(X)-
23.
Study of U-homomorphisms continuous on the right.
We shall nog½ suppose that g is additi,e. We shall say that the
mapping 9 ,from :; into ;5 is a U-homomorphism cont;nuous On
the right if it is continuous on the right in the pre,iously defined
sense, and if (AiUA..,.) .-- ? (A.,)U? (A.,.,) g,,bene½,er A, and A., ;.
Such. a mapping is clearly increasing.
196
GUSTAVE CHOQUET
23. I. 6eneral examples of U-homomorphisms continuous on
the right.
() Let E:. F and let be an additive class of subsets
of E.
(i) For every A c E, let ,qa be the class of those subsets of E
which are of the form (XU, A), where X a g. Then the mapping
X -- (XUA) from 8 onto . is a U-homomorphism continuous
on the right.
(ii) For every closed subset A of E, let , be the class of those
subsets of E which are of the form (XVIA), where X
Then the mapping X -- (Xf'IA) from g onto , has the desired
property.
() Let E F. If g is the class of all subsets of E, and if ,q
is the class of all closed subsets of E, then the mapping ?
from g onto .9 which is defined bY ?(A.).= A has the desired
property.
(T) Let x "..' (y) be a continuous mapping from a compact
space F into a Hausdorff space E. Then for every class g of
subsets of E, the mapping ?.. -' from 8, into the class of
all subsets of F has the desired property.
Indeed, for any A g let B ?(A)-- -'(A). Let V be .an
open neighborhood of B. For every point xa A there exists
an open neighborhood ' .
u of such that ,-(u)V If
U-----Ju, then. U is an open neighborhood of A such that
+-'(U) V, which proves the continuity on the right of ?.
() More generally, let E be an arbitrary topological space,
F a compact space, and A a closed subset of (E x F). For every
X c:E, let Y ?(X) be the set of those points y of F 'for which
(, y) a A for at least one a X. Then the mapping A-+(A) is
again a O-homomorphism which is continuous .on the right.
To these results there corresponds a reciprocal proposition
which shows, in an important special case, how every U-homo-
morphism which is continuous on th e right can be obtained.
Let g be an additive, hereditary class of conpact subsets of a
topological space E, and let Y 9(X) be a U-homomorphism,
continuous on the right, from g onto a class of conpact subsets
THEORY OF CAPACITIES
'197
of a Hausdori7 space F. Then titere exists in (E x F) a closed
subset A ,,hich satisfies the [ollogeing relation:
For e,ery X a g, ?(X) is the set o[ all points o[ F such that
{x, )a A for at least one x a X.
An equivalent statement is the following: if pr.(m) and
pr(m) denote the projections of a point m of A on E and
F, respectively, then (X) pr(pr'(X)).
We leave the verification of this proposition to the reader.
(s) Let y ...... ?(x) be a continuous mapping from E into F;
then the extention of 9 to an additive class g of subsets of E has
the desired property.. We have already used this example and
stated an important special case of it in 22, 2.
23. 2. Preservation of the class et(a "__ (1, b)) by the U-homo-
morphisms continuous on the right. Let E and F be two
topogical spaces, g and' two additive classes of subsets of E
and F, respectively, and let ? be a U-homomorphism, conti-
nuous on the right, from g into
If f is a capacity of order c%(a__: (_, b)) on , then the capa-
city e(X)--f(9(X)) on g is also of order et. This result is an
imnediate consequence of the fact that the definitions of the
classes a involve the operation U only.
24.
Study of Iq-homomorphisms continuous on the right.
Let us suppose that g. is multiplicati,e. We shall say that
the mapping ? from g into is a f'l-homomorphism
continuous on the right if it is continuous on the right, and i[
?(A, VI A) "?(A,) f'l ?(A) gehene,er A, and A are elements of g.
Such a mapping is ob,iously increasing.
24. i 6eneral examples of f-homomorphisms continuous on
the riõht.
(a) Under the conditions specified in example 23. i. (a), the
mappings X-- X U A and X- X fq A are f'l-homomorphisms,
continuous on the right, whenever g is multiplicative.
() Let E.-F.' If g is the class of all subsets of E, and if
is the class of all open subsets of E, then the mapping ? from
i98 aVSTZ. V COQV
onto defined by ?(A) k (interior of A) has the desired
property.
(7) The'mapping ?--+-' denned in example 23. t. (), has
the desired property. Thus, this mapping is both a U-and a
fl-honomorphism, continuous on the right, whenever g is
both additive and multiplicative.
(8) Let y = (x) be a continuous, one-tO-one mapping from
E into F, or more generally, let ? be continuous and such that
?(A, fl A) --- ?(A:)f'l (A.,_) whenever A, and. A. are elements of g,
Then the extension of. ? to g has the desired property.
(a) Suppose that E is a Hausdorff space and that every
element of g is compact.
(i) If F is the topological space of all compact subsets of
E, andif, foreveryA ag, wedefine?by?(A) t(A), where .;(A)
is the class of all compact subsets of A, then the mapping ? has
the desired property. For, on the one hand,
?(A, fl A.)... ?(A,) fq ?(A,)
and, on the other, the continuity on the right follows from the
definition of the classical topology of F.
(ii) Let I be any set of indices, and F the topological space E'.
If, for every A a g, we set B (A)- A , then the mapping
has the desired property.
24. 2. Paosra. It would be interesting to find a simple
method for the construction of every Iq-homomorphism,
continuous on the right, from the class g of all compact
subsets of a compact space E into the class . of all compact
subsets of another compact space F.
24. 3. Preservation of the classes t,,(.(t, b)) by the
f'l-homomorphisms continuous on the right. =. Since there is a
perfect analogy with the prøposition concerning the preservation
of the classes t by the U-homomorphisms (see 23. 2.), the
results will not be stated in detail.
24. 4. Study of other changes of variables. There are
other changes of .variable, such .as for instan. ce those which
transform a capacity of order t nto a capacity of order.,
THEORY OF CAPACITIES
].99
or conversely. They are of particular interest when the classes
and ff to xvhich they apply are classes of coinpact sets.
In this connection, it would' be interesting to find a simple
method for constructing every mapping 9 of the following types:
E and F are two compact spaces, g and
compact subsets of E and. F, respectively,
from 8 into ,9 which satisfies either
, the 'classes of all
and ? is a mapping
or
?(A, U A) - ?(A,)fq ?(A,.,)
?(A, fq A) -- 9(A,)U ?(A.).
The first of these two functions may be called an exponential
and the second a 1ogarithn. Both are decreasing; it is,
therefore, no longer possible to speak of their continuity on. the
right. In each particular case one should impose the type of
continuity which is the most suitable.
EXAMPLE OF AN EXPONENTIAL. "For a given E let T be an
auxiliary compact space, F a compact topological space of
continuous mappings from E into T, and A a compact subset
of T. For every compact subset X of E, we denote by Y "" ? (X)
the class of all continuous mappings from E into T which
belong to F and which map X into A.' "Then Y is compact,
and obviously satisfies the relation ?(X, U X) ?(X,) f 9 (X..,).
25.
Construction of alternating capacities of order 2.
Although the most interesting capacities to study are those
of order et= or , the fact that the capacities of 'order a
and b lead to a complete theory of capacitability induces us
to investigate the operations which lead to such capacities.
We shall study here an operation which leads to functions
which are alternating of order 2.
25. t. Study of the' 6reenian capacity by means of the Dirichlet
inteõral. Let D be a GreenJan domain of R". Let .'; be-the
set of absolutely continuous functions which are:non-negative
on D, zero on
Dirichlet integral
the boundary' of D,
and
possess a finite
f(9) - f (grad ?)" dx.
14
200
GUSTAVE CHOQUET
It can be shown that if , and % a , ?,
in a, and that
and ?, ?: are also
f(t,,-, ?,) + f(?, ,-, ?,,) .-,-. + f(%).
The n let K' be a compact subset of D. If we set
cap (K) .-- inf f(9) for all
it can be shown that, within .a constant factor, this capac. ity.is
precisely the Greenxan capacity Of K that we have stu&ed n
Chapter II. Let us show that cap(K) is an alternating capacity
of order et, (which we know already, but this new proof can be
extended to new cases).
The fact that .it is'increasing and continuous on the right is
immediate. Then let K, and K,. be two compacts of D and let
> 0. Let ,, ,, be two elements of J) such that:
f(?,) cap (K,).<Cs and ?,(x) t on K, (i - 1, 2).
We have therefore
Now
and
+ cap K, q-cap K,., 4- 2a.
(?, .-. ?. > !
on
(K,U K,)
(?, ,-, %)_ ! on (K, f'l K,).
It follows that
f(K, U K,) 4- f(K, 1'1 K,)<f(K,) -4- f(K,).
S nce ths nequahty s sufficient to obtain the most precise
results of the theory of capacitability, it is interesting to. try to
apply the above reasoning to a more general case.'
N. Aronszajn [] in his study of functional completion
and of exceptional sets associates a set function to each nor-
med space of- real functions on a given set E, in the follo-
wing way: Let II?tl be the norm on g. For each X cE we set
F(X)- infill i for all 9 which are ___i on X.
If there exists no ? which is ?_ t, on X, we set F (X)
In t'he case where ff is the linear space generated by the set
?.9 introduced above, the Greenian capacity of a compact.set X
is n fact the square of the expression F(X)corresponding to
THEORY OF CAPACITIES
20t
the norm iiq11- vf(?); but thi s difference is not trouble some
for the theory of capacity.
The following contains a theorem which leads to some gene-
ral cases where the above function F(X) is alternating of order 2.
25. 2. Alternating functions associated with a subvaluation on
ß n'
a lattice .- Let L be a lattice, a sublattice o[L such that each
a a L is ma/orated bt an a , and [ a real [unction on L' such.
that
f(a ,, b) -J- f(a b) f(a) + f(b).
We sa that f is a sub-valuation on L'.
When L' : L is a distributive lattice and' when i/s is an
increasing valuation. on L, we shall see (26. 4.) that [ alter-
nating of order ½: on L, relative to the operation .
If f is not an increasing valuation, this is no longer true in
general. However, we shall see how we can still associate to
each valuation and likewise to each sub-valuation on L' an
alternating function of order 2 on L, even if L and L' are not
distributive.
For each x a L, we set
cap (.) -- inf f(a) for all a such that x - a and a a L'.
Tzoa.. The [unction cap (.) is an alternating [unction.
of order 2 on L, relati,e to. the operation ,..
Proo[. That cap (x) is increasing is immediate; and the
inequality
cap (a,.b) q- cap (ab)<_ cap (a) q- cap (b)
is proved. exactly as in the case where [ is the Dirichlet integral.
EXAMPLES. Usually, the sub-valuation/will be a valuation
Here are some examples.
If D is a domain of R", we take L'= SS+ø* and take for L'
the set of real positive functions ? which are: continuous on D,
zero outside of a compact set, and Lipschitzian. If we set
(t,) SS+ denotes the cone of all positive and
on D which vanish outside of some compact.
upper semi-continuous functions
202
% when ?,(x)< %(x) for every x, L is a lattice and L' is
sublattice.
Let x, ?, ix/ be a continuous function. of x, ?, and of the
partial derivatives of the first order of ?, such that. fl (, 0, 0)d
(where ?. is the LebeSgue measure or any othe r fixed abso-
lutely continuous measure on D) has a' sense. We set
f(9) .... /oreach ?aL'.
'It is immediate that if ?, %, we have
f(t, ,-, ?,) -4- f(t,--, ?..,) -- f(?,) -4- f(t,).
The fact that this relation holds when ?and ? are arbitrary
is due to the following facts:
(a) the set of points of D where 9, =f-? 'is a denumerable
union of partial domains in each of which we have either
(b) the set of points of D where , --?a is the union of two
borelian sets A and B such that on A the functions ? and ?
have equal differentials, and B has Lebesgue measure zero.
It is often useful to notice that for each >0, and- for each.
neighborhood V of the support of any a L', there exists a
function ?' indefinitely differentiable, zero outside of v"With
iq- - ?'J <a and < .
These conclusions would no longer hold if ill the function.
some partial derivatives of ? of order 2 occurred,
Special cases.
(a) __ ? leads to the'norm of the spaces L .
(b) ]?., (grad ?) leads to the Dirichlet integral.
(c) - (t -4- grad ' )/" leads to the ½½ area of the graph of ?.
When ½1 is homogeneous of degree a with respect to and
i_, the function [cap (?)]'/ is honOgeneous of degree ] and, if
a > 1, the fact that v-- u '/" is then.an increas!ng and conc..ave
function implies that [cap()] '/ s alternating of order 2
whenever [() is .'_ 0 and alternating of order 2 (see 2]. 4.).
THEORY OF CAPACITIES 03
2!5. 3. Equilibrium. The definition given above of the
function cap (x) is more general, even in the setting of the
classical Greenian capacity, than the ordinary d6finition,
since it defines not only the capacity of the characteristic
functions of compacts, but also the capacity of any element
?SS
We. ould associate to every element ? a SS. a. sub-harmonic
function analogous to an equilibrium potent.al. We shall
show, in the general scheme introduced above, how we can
define such an equilibrium in 'very general cases.
Let us use the notations introduced in the above theorem.
For each a0 a L, let L(a0) be the set of elements a of L such
that ao- a and cap (a)--cap (a0). L(a0) is a sub-lattice of L;
in.fact, if. a,, a, a L(a0), we have
so that since cap (x) is increasing, we have
cap (a, aa) -- cap (ao),
and hence
a, -, a a L(a0).
On the other hand,
cap (a, aa) -4- cap (a, aa) < cap (a') -4- cap (a);
hence
(a,
and since cap (x) is increasing, it follows that
a, ,- a a L(ao).
The lattice L(a0) possesses a smallest element, which is ao;
it can have 'only one largest element; when the latter exists,
we shall denote.it aq; it is the equilibrium element associated
with ao.
A case where aq always exists whenever ao is su'ch that
L(ao) is bounded above is when
(a)
L ,,,.,, L';
each subset of L bounded above possesses an upper bound;
[(x) is lower semi-continuous on the left, which means
20: GUSTAVE CHOQUET
that for each subset (a) of L, filtering on the right, and having
an"upper bound a,o, we have
f (ao) -< lim inf f (a,).
This semi-continuity occurs, for example, when f(?) is the
) which is > 0 and has,
integral on D of a function (I) x, ?, i)x "'
in a certain sense, a convex indicatrix when considered as a
function of i). (Example: (I)
'- grad ? or (I) --
(! q- grad
It can happen that for some ao a L, L(a0) is not bounded
above, but that by introducing a convenient notion of excep-
tional set, L(a0) possesses a quasi-upper bound. This happens
for example in the classical potential theory.
26.
Examples of alternating capacities of order g=.
In all of the following examples, the capacities under
consideration are always tacitly assumed to be defined on the
class g -- (E) of all compact subsets of the space E in question
unless otherwise indicated. We shall give here only examples
of capacities of order' et. Let us notice here that many
capacities which occur naturally in analysis are obtained from
Radon measures by a small number of operations such as
.U' C,. max, min, and that in general, the capacities obtained
m ths way either fail to be of any order g or ,b or they
are of order cor
26. . Alternating family of elements-of a commutative ordered
group: Let G be a commutative ordered group, and. I
a fimte set. Every function, alternating of order oc, which
is defined on the class 8
values are in G is called
merits of G. Thus, if x
be non-positive.
used before,
2' of all subsets of I and whose
an. alternating [amily (x)j, of ele-
f(J), all the Vf are supposed to
Let us set, conforming to a notation already
V[(I a); {i,l,, ]-
(J I, with J =/= p)(Xj .2 0).
oSy o cCT.S 205
By an already familiar computation we deduce from these
relations the following:
KfiJ:
If x !s n.ot de.fined, an .arbitr. ary .value suc. h that ), is non-
negative s assigned to X; ths assignment s. always' possible,
ConVersely, it is easily verified that every family x which is
defined by equalities of this form with numbers ),>0 is
indeed an alternating family.
ExarLS. If I contains two elements t, 2, then every
alternating family on I is of the following form:
26. 2..Operation ,, sup ,, in a commutative lattice group (').
Let G be a commutative lattice group and I any set. Also, let
i -- x be a mapping ? from I into G.
Set/(X)- sup (x) for every finite X I. The function [
x
is thus defined on the additive class 80 of finite subsets of I.
We wish to prove that the V½ are non-positive and more
precisely, that
V (x; I A )y-- inf If(x),
inf [lf(A)i].
It is equivalent to prove that for arbitrary elements x, % of G
(p--i, 2, ...), we have
V(x; l%)..-inf(x, a) a
where
a--lnf % .
We recall the following identity: inf (u, ,) q-sup (u, ,) '" u q- ,.
It follows that V, (x; a,).--- x sup (x,
The general formula follows from this one by induction: the
proof is entirely analogous to that of 14. 5. for functions which
are both alternating and monotone of order 2. Thus, we can
state that the operatiøn ( sup ) in a commutative lattice group
is an alternating function of order infinity.
(t) A commutative lattice group G is an ordered group such that any two ele-
ments X and Xa of G always have a least upper bound, sup(Xt, Xa), and a greatest
lower bound, inf(Xt, X2), sometimes denoted by X -Xa and X X 2, respectively.
206 srv cioQur
For the operation inf there .is a formula which is the dual of
the preceding one; hence, the Vf will be non-negative. Thus,
if for every X I we set
co(X) .... [sup (a:;) --inf (a:)],
ix
the oscillation to(X) is an alternating function of order infinity.
If G is. in addition a complete lattice ('), these results may be
extended to additive classes g of subsets X of I such that every
.?(X) is bounded from above (and also bounded from below if
o(X) is being considered).
APPLICATION. :: Let. ?(x) be a real-valued continuous func-
tion on a topological space E. For every X c E we shall
denote by f(X) and co(X) the least upper bound and the oscilla-
tion of ? on X. These two functions are alternating capaci-
ties of order c on each additive class of .subse. ts of E (on
which they are assumed to be finite, for smphficahon). When
?(x) is only 7pper semicontinuous on E, f(X) only is a capacity
of order ct
Exaur,s.- If 8(X) denotes the diameter of a compact
subset X of the real line, then since (X) is the oscillation of
the function ..z on X, this diameter is a capacity of order ct= of X.
(It should be remarked that if one wants to assign a value to 8(½),
this value should be m).
On the other hand, the diameter of a compact set X in an
arbitrary metric space E is not of order c%. This diameter is
equal to the maximum of a function which is defined on E
and not on E. We therefore have only 8(X)=--f(X ) where [
is a capacity of order ct on .(E).
26. 3. 6eneralization: valuation on a distributive lattice. ,..
Let L be a distributi,e lattice and [ a mapping from L into a
commutati½,e ordered group G. We shall say that f is a ,aluation, if
f(a,- o) + f(a o) f(,) + f(
26. 4. Tusoua. . If f i, an increasing ½,aluation (that is, if
(a - b)=. [(a) f(b)), and if me set g(X) = f(sup X) [or e,ery
{16) A lattice O is said to be complete if and only if every subset of G which is
bounded from above possesses a least upper bound (and likewise for the greatest
lower bound).
ß .oeY o cexcms 207
finite subset X of A, then the [unction g(X), ,hich is defined on
the additi,e class g o[ all finite subsets o[ L, is alternating o[
order , and
V(X; IA, l)-inf(g(X), lg(&)i) infig(&).
An analogous statement holds in the case where L is a complete
distributive lattice and where g is the class of all bounded
subsets of L.
CoaorranY. .. With the same notations, the [unction f(x) is
alternating o[ order on the ordered semi-group L with the ope-
ration sup.
26. 5. Examples of such valuations.
(i) The dimension of a variety in projeetive. geometry or
in Von Neumann's continuous dimensional projectire geometry.
(ii) For L we take the set of all positive integers, ordered
by the relation a - b if b is a multiple of a and we set:
f(x) --, Log (x)
g(X) -- Log [1.e.m.(X)]
for every X L,
with X finite.
26. 6. Non-negative. Radon measures. If E is a locally
comp.act space, a function f defined on .(E) defines a non-
negative Radon measure if and only if
(i)
(ii)
(iii)
(iv)
f is finite for every K .(E).
f() .... 0
f is increasing and continuous on the right.
f(K, U K.) q- f(K, 13 K) f(K,) q- f(K).
These conditions are equivalent to stating that [ is a capacity
on ..'5;(E)-of orders a. and 'l.b which is finite and such that
f() ' 0
More generally, if E is any Hausdorff .space, any function f
which is defined on an additive and hereditary class g of
compact subsets of E, and which satisfies the conditions (i),
(ii), (iii), (iv), will be called a generalized non-negati,e Radon
measure on g. Here agmn, these conditions are equivalent to
the statement that f is a capacity on g of orders cx and
which is finite and such that f()- 0.
We further remark that, since the class g is rich (see
208 a,Jssv. c,,o,
Chapter v, 7. 3.) the extension of f to the class ,'.(E) of all
compact subsets of E is still of order et by virtue of Theorem
7. 0. of Chapter xv. Since, on the other hand, ;(E)is
G-separable (see definition 8. 5.), this extension is also of
order d/b. by Theorem 8. . of Chapter xv. Thus, this
extension to (E) is a capacity of order et and A!b.. such that
f(½)- 0. But.it may happen that for this extension
.f(K) -+- . for certain compact sets K.
Let us show that i[f is any [unction hich is 'defined on an
additi,e, hereditary class g of compact subsets o[ E, and which
satisfies conditions (i), (if), and'(iii), the condition t iv) is equi-
,alent to the lollowing condition:
(iv') f(K.: U K) f(K,) +/(K.), and this inequality becomes
an equaat!t he'n%er K, 0 K O (K, and K are elements
of
Indeed, since f 0 and f (')- 0, (iv)implies (iv'). Conver-
sely, let us suppose that (iv') is satisfied. We wish to show
that, if K, and K, are elements of $, then
f(K, U K) -!- f(K, rl K,,)-- f(K,) -I- f(K).
If K,f'l K , the desired relation obviously holds. If
K, fl K --/= ½, let a be an arbitrary non-negative number, and
let V be a compact neighborhood of (K, fq K)in (K,U K.)
such that
f(v)--f(K,nK.,)
Set(K,.. r)--k, (i
are disjoint and
((K, rlK,) U
2). The compact sets k and (K,0K.)
k,) K, (V U k,..) (i
t, 2).
Hence, by virtue of property (iv'),
f(k,) + f(K, rl K:) _<___ f(K,) f(k,) + f(V),
and
f(k,) + f(k,) + f(K, f"l K,) f(K, U K) .
Therefore,
f(k,) +
+ f(v).
ß f(K,)q-f(K,)- f(k,)+ f(k,.)' + 2f(K, r'l K.:) + -½,
where 0 :< q 2,
and f(K, U K,)
THEORY OF CAPACITIES
209
where
Thus
where
f(K, U K..) + f(K, 1. K) -- f(K,) + f(Kd +
0 :-" 2.
Generalization. There exist functions of compact sets
which are closely analogous to the generalized Radon measures
but which are not continuous on the right. For instance,
the linear measure of Caratheodory, defined on the class of all
compact subsets of the Euclidean plane, is such a function.
In this connection, it is of interest to introduce the following
definition.
We shall call any function f, defined on an additi,e hereditary
class $ of compact subsets of a Hausdor# space E a Caratheodory
measure if for e,ery element K of its restriction to the class
of all compact subsets of K is a non-negati,e Radon measure
Or Io
26. 7. Newtonian or 6reenian capacity. If E is a domain in
the Euclidean space R ", or more generally, if E is a conformal or
locally Euclidean space which possesses a Green's function (see
Brelot and Choquet It]), .then the capacity of a compact
subset K E with respect to this Green's function is of order
ct.. We have studied these capacities in detail in Chapter II.
26. 8. Fundamental scheme of the capacities of order
Let E and F be two sets (without topologies), A a subset of
(E X F), and a non-negative additive function defined on
a ring (') 5 of subsets of F. For every subset X of E, let
?(X) be the projection on F of the set of those points of A
whose projection on E lies in X. In other words
? (X) pr [A fq (X x F)i.
The mapping X .-f.?(X) is a U-homomorphism.
Let g be an addrove class of subsets of E such that 9 (g)c ..
(7) A set xvhich is closed 'under finite union and under difference, hence also
under finite intersection.
2.'1.0
GUSTAVE CHOQUET
The function . is alternating
4. 5.). Hence if we set
of. order oe on (see Chapter III,
f(X) .. ((X))
for every
the function f is alternating of order c on g.
For instance, if E is a Hausdorff space, F a locally compact
space, A' a closed subset of (E x F), . a non-negative Radon
m. eas.ur.e on F, and 80 the class ..(.E) of all compact subsets of.E,
'men t s easy to show that ? (X) s closed for every X a g, henc e
measurable with respect to the measure ,., and the preceding
definition is applicable. If one can show in addition that/(X) is
continuous on the right, one can then state that/(X) is a capa-
city of order a. This case
set A is compact, or more
every compact XE.
will be realized, for instance, if the
generally, if ?(X) is compact for
We shall say that f is the function (or the capacity) obtained
by the fundamental scheme (E, F, A, .).
It is clear that in this scheme the additive. function I.. could b e
replaced by any alternating function of order oc, but this
generalization is not of great interest; on the other hand, we
shall see that the importance of this scheme 'lies in the fact
that it provides a canonical representation of every capacity
of order on E, provided only that this capacity satisfies
some conditions of regularity.
and let
xX.
26. 9. Game of ,, Heads or tails ,,. Let E be a finite set of
throws in a game of ( heads or tails . For every K E, let
f(K) be the probability of the event that tails occurs at least
once on K. The function f(K) may be obtained by the following
scheme: let F 2 z be the class of all subsets of E (including
A c- (E X F) be the set of all points (x, X) such that
If is the measure on F defined by the condition that the
measure of each of the 2 points of F be 1/2 , then f is the
function obtained by the
alternating of order cx .
We remark that f(K)
scheme (E, F, A, ).
Thus [ is
depends only on the number of
elements of K; if that number is n, then f(K)--?(n).
Now if X, A,,..., Ap are subsets of E which are mutually
THEORY OF CAPACITIES 2tt
disjoint, with cardinal numbers n, a,,..., a,,
then we have obviously
respectively,
V(X; IA, I)r--'X7(n; lal),
and this equality shows that ? is . function of n which is alter-
nating of order infinity. This can be verified by using the
following explicit expression of ,: ½(n)'" (.2-).
26. 10. 6eometrical probabilit7. Let E be a plane, D a
line in the plane, and ?. a non-negative Radon measure on D;
for every compact subset K of E, let/(K) be the .-measure
of the orthogonal Projection of K on D. Then œ(K) is obvi-
ously a capacity of order a. on 3r(E). (As an analogous
example, we can consider the { angle /(K) under which a
compact set K, assumed to be contained in (E 0), is
seen
from a fixed point 0 of the plane.)
From this remark we might deduce that the measure (here
assumed to be the classical invariant measure) of the set of all
lines of the plane which meet a compact set K is a capacity of
order %. But it is more convenient and more interesting to
prove this by means 'of the fundamental scheme as follows.
Let F be the topological space (which is locally compact)
of all lines D of the plane; let be the invariant classical
measure on F, and A the closed subset of (E X F) which consists
of the pairs (x, D) for which x a D.
The function œ(x) vhich is obtained by means of the scheme
(E, F, A, ) is obviously the measure of the lines D which meet
the compact set D. (If K is convex, then f(K) is, moreover,
equal to twice the length of the boundary curve of K.)
Now let us consider only those compact sets K which are
fixed circle in. If we
set p(K)'
, /(K), then the
f(r)
contained in a
function p(K) represents the probability of the event that a line
which meets [' also meets K. As in the preceding example we
have here exhibited a probability which is a capacity of ordor et.
We shall return to this investigation .in the last chapter.
26. tl. Let be a non-negative Radon measure defined on a
compact metric space E, and let h(u, m), (u 0, m E), be a
2i2
GUSTAVE CHOQUET
continuous function of the point (u, m), which is decreasing in
u for every m.
For every compact subset X of E, set
: h(u,,,, m)
f(x)
where U m denotes the distance from m to X.
We shal! show .that œ is a capacity of order et on :,(E).
Indeed, [ s obtained by means of the fundamental scheme
[E, .(E), As, a], where :.(E) denotes the compact topological
space of all closed subsets of E, Az is the closed subset of all
points (x, X) of (E X .(E)) such that x a X, and is a non-
negative Radon measure on that subset B of J(E) which
6onsists of all closed solid spheres B(m, u) of E, with ,a defined
by the elementary measure dh. d(m).
For every X a (E) the class of the compact sets which meet
X has -measure zero with the exception of those which are
closed spheres; and the spheres B(m, u) which meet X are
those for which u > Urn. Hence the result.
26. t2. Harmonic measure. Let E be a Greeninn domain,
and for every m a E and every compact subset X of E, let
h(X, m) be the harmonic measure of X with respect to the
point m for the domain (E X). (When m a X, we shall set
h ' l, by definition.) We have already used the fact that this
function is quasi-everywhere equal to the equilibrium potential
of X for the Green's function of E. (See it. 2.) Moreover, we
have shown (see 7. 5.) that the equilibrium potential .of X
considered as a function of X, has all its differences (V)
non-positive. Thus h(X, m)is an alternating function of X,
of order , for every m. It is continuous on the right. This
fact is obvious if m, X, and, if m a X, then h(X, m)-
hence, we have also h(X', m)- i' for X' X. Thus h is
indeed a capacity of order a. '
More general capacities of order may be derived from
this one by setting/(X) , (h(X, m)d.(m), where is a non-
ß
negative Radon measure on E of finite total mass.
We have given this example immediately after example
26. it. because of their great similarity.
THEORY OF CAPACITIES
2i3
26. 3. Construction of (antor-Minkowski and reõularization
of a capacity. Let E be a metric space such that every
closed sphere in it is compact. For every compact subset K
of.E, and for every number p 0,.let K(,) be .the set of all
Points of E whose distance fro m K s at most p
The mapping g- K(,)is a -homomorphism, continuous
on the right, from 3(E)into 3(E)..Hence, if g s a capacity o.f
order t= on 3(E), then the same s true for e, where fe
d. etned Moreover, decreases to gasa
hmt, as
For example, in E- R", may denote the Euclidean
measure of K.
This construction may be used to show that e,erg capacity g
of order eL, on 3{(E) is the limit of a decreasing sequence of
capacities o[ order a. on {E), each o[ gchich is a continuous
function o[ its ,ariable X.
For simplification, let us suppose 'that g.. 0. Let ?(u)
be a real-valued function of the real variable u, defined and
continuous on [0, ], decreasing, vanishing at x- l, and such
that fo ?du- _. For every k > 0, we set
gx(K) :." . .fo f.(K)k?(),u)du.
We may also write
g(K) .,-- fo f/x(K)?(t) dt,
which shows that g(K) is a decreasing function of X. The
function g(K) is on the other hand, clearly an alternating
function of.order o of K since this is the case for f(K) for every
u. And since for 0 t , [t/ (K) tends uniformly to g (K) as
), > or, it follows that-x(K)-- g(K).
It remains to show that gx(K) is, for every k, a continuous
function of K considered as an element of the classical topo-
logical space of the compact subsets of E. If we use the
classical metric for this space, the distances of any point o!
E to K, and' K,. differ by at most whenever (K,, K)_<
which implies that K, () K. (p q- ) and K. () K, (p -).
Thus /,(K,)</,+(K.) and/(K) f.(K,), so that
2t4
GUSTAVE CHOQUET
But
[f..(K)-. -f.(K.)]kg(ku) du
,,,,,, f.(K)X[?(k(u .... ))
so that
Let. M--..f(K) for
where s0 > 0.
For every s < So, we have
and an
Thus
(x,,)]
gx(K,) <fMX[(X(u., a)) (x>11&-Mfo X
_ . --- q (t) dt,
analogous inequality by interchanging K, and K.
}gx (K,).., gx (K,)I
ß ? (t) dt for every s < 0,
which shows that gz(K) is continuous.
Note that any alternate capacity on
continuous on the topological space 3(E). We have just
shown that it is a decreasing limit of continuous capacities of
the same order.
(E) is upper seni-
26. 14. Elementary capacities of order a.. Let E be a
Hausdorff space and œ a.capacity on P,(E! which is sub-additive
and whose range contains at most the; alues 0 and
Every element A a3i',(E) such that /(A)= 0 has an open
neighborhood o such that, for every compact Xo we have
/(X) 0. Let ,Q be the union of the open sets
Every compact B P. is covered by a finite' family (o) of
these open sets ½o; therefore that compact B is the union of a
finite number of subcompacts each of which is contained
in one of the ½o (see, for instance, 17.4. in Chapter iv).
ß Therefore œ(B) 0...In other .w. ords, for every X ?(E),
the necessary'and suffiment condmon that œ(X)- 0 s that
XO.. Let T 0..
0 if X fi T.,.-- p.
We have /(X) -- 11 if X Iq, T =/= p'.
THEORY OF CAPACITIES
Consersely, if T is any non-.empty closed subset of E, the func-
tion œT(X) defined by the preceding relations is obviously a
subadditive capacity on ,(E).
We shall prove in Chapter v as'a special case of a general
'theorem that every fT(X) is a capacity of order et© and that
these capacities are the extremal elements of the convex cone
of positive. capacities of order et on .(E).
The function fT(X) will be called the elementarg capacitg
(with index T) of order
27. Examples of capacities which are monotone of
We shall give here fewer examples tha n for
of order eta, at first because nonotone capacities do
as often as alternating capacities and also because
to be less useful. '
order t,, .
capacities
not occur
they seem
27. i'. Every non-negative additive set function is. nonotone
of order .. Thus each non-negative Radon measure on a
locally compact space E is a capacity of order ,itto on 3,(E).
27. 2. The fundamental Scheme of alternating capacities
is replaced here by a scheme that we shall indicate in a special
CaSe.
Let E be a locally coinpact space, F the topological space
of its compact subsets, and a non-negative Radon neasure
on F. If, for every K c,E, we set f(K)-- (3;.(K)) where
denotes the subset of F consisting of all. the subconpacts of K,
then f(K) is a capacity of order
The interest of this schem e lies in the fact that it leads to a
canonical representation of all positive capacities of order
on J'(E), as we shall see in Chapter
27.3. Let ,a be a non-negative Radon neasure on a locally
compact space E, and let h(P, Q) be a non-negative continuous
real-valu. ed function of the couple (P, Q), or more-generally
a Baire function (with, if necessary, the restriction that
P-%Q).
The function/(K) =
ord e r, il;b
..! h(P, Q)d (P)d, (Q)is a capacity of
on ;q'i'(E), for the napping K-----K ø' is a FI-holnomor-
15
2!6 GUSTAVE CHOQUET
phism continuous on the right, and h(P, Q)d(P)dl(Q)defines
a Radon measure on E a (with possible value q-c).
Let us remark that œ(K) can be interpreted as the energy of
the restriction of to K for the kernel h(P, Q).
There are analogous statements for a function h of n variables
defined on E .
27. 4. On E.-. R", if we define f(K) to be the Euclidean
measure of the set of centers of circles of radius contained
in the compact K, f is of order ,.,1.
27. 5. On E '-' R a, we set f(K).--.h(p(K)), where p(K)
denotes the radius of the largest sphere with center 0 contained
in K, and h(u) a function of the real variable u...=_. 0 which is
non-decreasing and continuous on the right.
The function œ can be obtained by the scheme of 27.2. above
where is th e Radon measure defined by dh(u) on the set of
spheres with center 0. Then [ is a capacity of order 1.
27.6. Let E be a finite set of throws in a game of heads or
tails. For every KE, let/(K) be the probability that tails
occur nowhere except possibly on K.
This probability is within a constant the conjugate function
of the probability that tails occurs at least once on K.
Iris then oforder,,l. If'K ..... n and E = a, then/(K) - 2/2;
and it can be verified that/(K} is a totally.monotone function
of n in th.e classical sense.
27. 7. Elementary capacities of orders ,lt.,=. Let E be a
Hausdorff space and [ a capacity on (E) which is of'order
and whose range contains at most the values 0 and . If
/(X,) - f(X) -- , we hav also/(X., fl X,) .. $ and unless [
we have X f'l X, . Therefore the set of elements X a (E) for
which/(X) = does not contain and is multiplicative.
Let T be the non-empty intersection of those compacts; as T
is also the limit of that decreasing filtering set of compacts and
since f is continuous on the right, we have. f(.T)
In other in X)" , t s and
words, order that œ( necessary
sufficient that TX..
THEORY OF CAPACITIES
2t7.
Conversely, for every compact T E, let
if TX.
if T½ X,
It is obvious that/(X) satisfies the identity:
œ(x, n x) = f(x,).f(xd.
It follows from this (and it will be a particular case of a theo-
rem of Chapter v) that f(X) is of order vl,'b, and that these
capacities are the extreme elements of the convex cone of
positive capacities of order A. on (E).
The function f(X) will be called the elementary capacity
(,ith index T) of order
CHAPTER VI
GAPAGITABILITY. FUNDAMENTAL THEOREMS.
classes ,,. or ,llb.. In
terminology we shall
every class &;. of sets.
28. Operations on capacitable sets for capacities of order et.. ..
In this chapter we shall study the invariance of capacitability
Under the operations of denumerable union and intersection,
as well as capacitability of analytic sets. We shall see that we
can obtain subtantia] results for capacities which satisfy suffi-
ciently strict inequalities, for example, those which define the
order to avoid some complications of
suppose always that o is an element of
28. [. THEOREM. Let l; be an additive and rich set of
subsets o[ a topological space E, and let f be a capacity of order a.
(, b)) on g.
Each finite union of f-capacitable sets of capacity
is also f-capacitable.
'(ii) If
of E ,e ha½,e f(oo,,)- f(Uo,,) (for
ß pa t) th
zs com c , en
(a) f is of order 'a,., ; n other ,ords, *
ß f (A,,)
f is such that for each increasing sequence .1o, of open sets
'example, if each element of
each increasi.ng sequence
and
( b) each denumerable
of sets k, cE
z is also f-capacitable.
-..-f*(UA,) for
such that if(A.)5/= :;
of capacitable sets of capacity
Proo[. Notice that if f is of or. der a,.(n _ '), f'is also of
order et:. On the other hand, the nequahty.vMch defines the
class et,, is highly analogous to the inequality
f(A, nA,) f(a, + [f(A,) ...... f(a,],
TItEORY OF CAPACITIES
2i9
which is satisfied for the class e,. Thus,. in order to simplify
the notations, we shall give the proof only for the .class et.
We recall first that, when f is of order % and is additive and
rich, by virtue of the inequality in 4;. 3. and by Lemma f7.9.,
WC hav
f*( U A.,) .... f*( U a,) E (if(A,)
where a A E for each i.
.if(a,)),
and by remarking that [.J A .. A,,
f(0a) -- f*(A.) .< E +... + {vhere
f*(A,,) f*([.J A,,).
Otherwise for each s > 0 and'for each n there exists an open
set o such that A,, (o,, and /(con) -/*(A,)<( ]2 .
We have, therefore, by applying the inequality 28. 2. above,
that
[.JA, ), we have a fortiori
lim if(a,.)_<___ [* ([.J A,) __ lim
Now if we set Q = [.J L),, : [.J co,,, we know by hypothesis
that f(0.,)-- [(C)). Therefore, f(O)lim if(A,)q- , and, since
if(An) + .
Since s is arbitrarY, ,ve have limf*(A,)= f*( A',).
Proof of (i). It is obviously suificient to prove the theorem
for the union of two sets A, and A. Moreover, if one of these
sets, say A,, has a capacity f(A,)-k-o e, the set A, o A,. has
an interior capacity equal to q-c; therefore, it is capacitable.
We shall suppose therefore that /(A,). and f(A) are finite.
For each s > 0, there exists a set Xg and an open set
o E such that X A o and f(½o).. f(X) < (for i.-- 1, 2).
We have therefore, by applying the above inequality 28.2.,
f(, U to). f(X, U X,). [f(to,) .... f(X,)] + [f(o,) ..... f(X.)] < 2z.
( and
Since (X, U.X,)(A, U A.)(o, U o) and (X, U X.) a o,
since (o, U co.) is open, the set (A, U A,) is capacitable.
Proof of (ii). The proof of (a) Will be given ast. Let !A,,
be an increasing sequence of subsets of E such that/*(A,.) :/= .c
for every n.
If for n--no we have f (A,,)= +,e, it is obvious
220
GUSTAVE (IHOQUIT
The proof of (b) will be given next. Let A I be an arbi-
trary sequence of capecitable 'sets such that /(A)=/=.
Let B: A,.'We have JB: JAa, ana moreover, each B
is capacitable according to the first Part of the theorem. Since
the sequence B is increasing we have
lim f*(B) '- lim f,(B) = f*( B).
On the other hand we. have
lim f,(B) /',(U B )
Hence, f.(kJ B,)<f,( kJ B); the eapaeitability of (kJ B,) fol-
lows from this inequality.
We sh.al! now. show that if each element of g is compact,
the cond,t, on l,m f(to.) [ (.J to.) is-satisfied.
We have at once that limf(to.) f(), .where
On the other hand, if [() < 4' , for each a < 0 there exists
a compact K a g such that [(to)- ./(K,) < t. Now K
therefore, since K is compact and since the sequence o. is
increasing, there exists an n---n such that K,t%. It
follows that [(o) f(o..) < a. Therefore, f(o)lim/(ø.);henc e
the equality.
In the case where [() q- , the proof is Similar to that
given.
We remark that this result about open sets is valid for any
capacity on a class g of compacts.
28. 3. CO aOLLaaY. Let g be an additive and hereditarg
set o[ compacts o[ E.
If f is a cape_city of order et.(a,> (, b)) on g with f>--,
each denumerable "'union o[ capecitable sets is capecitable, and
[or each increasing sequence o[ sets A.E, we have
lim f*(A) -- f*(kJA., ) .
ff f is of order then for arbitrary finite or infinite sequences
of subsets (A.) and (a.) of E, with a A for each. n, ,e have
f*(U &)" f*(LJ a,) E(f*(&)-f (a)).
THEORY OF CAPACITIES 22i
29. /i capacitable class o! sets. ,- We shall introduce first
a convement terminology.
'29. 1. Dr. lqNITION. ' Let g be a class o[subsets o[a set E. We
shall let g denote the class o[ sets A E where A is a denumerable,
union o[ elements of g.
We shall let $o denote the class o[ sets A E ,here A is a denu-
metable intersection of elements of go. .
We want to show, that under certain hypotheses each
element of ga is capacitable. We cannot use for the proof the
fact that each denumerable intersection of capacitable sets is
capacitable, for this fact is already false for finite intersections
as we shall show later. We will therefore have to use in a
precise way the fac t that g is constructed from elements of g.
the set g satisfying in addition certain restrictions.
29. 2. T.oa.M. I[ g, additi,e and denumerably mult-
plicati,e, is such that, [or each decreasing sequence I A, I o[ elements
of g and e,ery neighborhood V of a (q A, gee ha½,e A V [or n
su[ficiently large, and i[ [ is o[ order a,,, each element o[ g is
f-capacitable.
Proo[. Let A a g;. Then A- A, where A a g ;in other
words, A --- A where
suppose, since g is
We can always
ncreases wth p.
Set f*(A).--1. If
additive,
that A
1 - oo, we have also
and A is capacitable. Otherwise it is finite or equal to q- .
We shall give th e proof in the case in which I is finite; the case
in which l--q-c could be treated in an entirely analogous
manner.
(Besides, the case in which l. q- c can always be reduced
to the case where 1 is finite by replacing œ by g --e -y. The
functio n ( .. e -) is continuous and strictly increasing; hence if [
is a capacity of order et,, , g is also. Furthermore,/-capa-
citability is equivalent to g-capacitability.)
Let e be any positive number. The set a'--AFIA is
increasing with p, and we have A- U aa.
f is of order et,,, we have f*(A)- lim
The efo '
r re, since
222 GUSTAVE GHOQUET
Therefore there exists an index p, say p, such that
f*(A), f*(aq') if.
Suppose that the sets a, have been defined for each i<n
in such a way that for each i, [*(a?) is finite and that a, c-A.
Set a+, . a fq A.,. This set is increasing with p, and we
have a- :- U a+,, from which it follows that
f*(a.) ..--,. lim f*(a.
There exists therefore an index p,
say p+,,
such that
f (a,+, ,)< 2,,% , ß
If we add the first n inequalities thus obtained, we get
29. 3. f*(A)
<
The a constitute a decreasing sequence of sets, all contained
in A. Set a ') a,,. We can also write a A f'l [A].
Now A, cA,, so that ' A cA; hence, a (' A.
If we set'B,;. '..'..'.. [')A% the B, constitute a decreasing sequence
!
of elements of g and a -- ')B. Now a is again an element
of 8; therefore, according to the continuity on the right of
œ and the given hypothesis on the mode of convergence of
decreasing sequences of elements of g, we have
- lira f(Ba).
Since a a]"Ba, we have also [(m)..--lim f*(a],). The above
inequality 29. 3. becomes/*(A) [(a).< . Since a a g, we
have therefore f*(A)L(A ) for each s. lienee/*(A)---f,(A).
29. 4. COIOLLAaY.... I[ g is an additi,e and hereditary set
o[ compacts o[ E, and i[ [ is o[ arbitrary order et on ; geith
[>- , each element o[ g** is [-capacitable.
In fact, according to the Corollary 28.3. of. Theorem 28. ., [
is then of order a.,, a and on the other hand, snce each element
THEORY OF CAPACITIES
223
of.g is compact, each decreasing sequence of elements of g
satisfies the exact conditions of Theorem 2'9. 2. Therefore,
this theorem can be applied,
Notice that in this case each element of g, is a Ko. But
it is not true that each K, of E is always/-capacitable.'We can
indeed construct examples where there are some compacts
of E Which are not/-capacitable, even if f is of order
The following is rather instructive as an example. Let E
be 'the Euclidean plane. R'", g. the set of com. pacts K of .the pl. ane
such that K is contained n a finite umon of strmght hues
parallel to a given fixed line 5. For each Kag we set
f(K) -- linear measure of the projection of K on A.
It is immediate that [ is continuous on the right on g and
alternating of order ex; on the other hand, 8 is additive and
hereditary.
Now for each cronpact KE we have œ*(K).. linear
measure of the projection of .K on A; and if K is such.that
each intersection of K with a hne parallel to 5 has a zero hnear
measure, we have f,(K)'-- 0.
For example each arc of a circle is non-eapaeitable for [.
Here the elements of go are the denumerable unions of sets
each of which is located on a line parallel to A and .is any K,
on such a line.
30. 6apacitability of K-borelian and K-analytic sets. We
shall extend Corollary 29. 4. to the K-borelian and K-analytic
sets.
30. 1. THEOREM. _If g is an additi,e and hereditary class o[
compacts of a Hausdorf[ space E, and i[ f is o[ arbitrary order
a on g and [>. , any K-analytic set A o[ E is f-capa-
citable in each o[ the follogeing t,o cases.
(i) A B mhere B a g, (example: A is an element of the borelian
field generated by
(ii) A o vhere o is a completely regular open set; and in
addition 3:t(A);, that is, each compact K A is an element
of
_Proo[2 In each of the two cases considered, A is such that
each compact K contained in A is an element of g, and hence
' f p it bl
is -ca ac a e.
224 GUSTAVE CHOQUET
Therefore, according to Theorem i6. 3. it is sufficient, in
order to show the f-capacitability of A, to prove that A s capa-
citable for the extention f of f to the set (E) of all compacts
of E.
Now g being additive and rich and 3;(E) being additive'
(E is Hausdorff), this extension [ is, according to .Theorem
7. 0. of class
Thus Theorem 30. . will be established if it is proved in the
simpler case where 8 (E).
We shall now simplify case (ii). It is sufficient to remark
that, since A is contained in a completely regular open set, we
can apply the method explained in Example 20.3. to reduce the
problem immediately to the case where the space E is compact.
In short, the two cases (i) and (ii) are both reduced to the
following simpler case: A is contained in a K of E and g 34 (E).
Now according to Theorem 5. . there exists a compact space
F and a set F E x F such. that F is a K, and such that its
projection on E is identical with.A.
Let us designate then by g the capacity defined on. (E x F)
by the equality g(X)-/(prsX), where (priX) means the
projection of the compact X on E.
According to 22.2 and.23. 2. in Chapter v, the capacity g is of
order ct,; since in addition g>--, according to Corollary
29. 4., F is g-capacitable. Therefore according to Theorem
22. 3. in Chapter v, its Projection A on E is /-capacitable.
30. 2. COROLLARY. I[ E is a space ,hich is homeomorphic
to a borelian or analyt]c subset (in the classical sense) o[ a sep a-
rable complete metric space, and i[ [ is a capacity > , defined
on the set 3(E) o[ the compacts o[ E and o[ arbitrary order eta,
each borelian or analytic (in.the classical sense)subset A o[
E is f-capacitable.
In' fact, Theorem 30. I. is applicable to A since A is contained
in the open set E which is completely regular, and since A
is K-analytic (according to the classical theory A is the continu-
ous image of the set of irrational numbers of [0, 1], which is a
K).
3i. (lapacitability for the capacities which are only subaddi-
tive. We shall no, construct an e:rample of a capacity
f __ O, sub-additive, defined on the set o[all compacts o[ the plane
THEORY OF CAPACITIES 225
E R ', and for vhich there exists a closed subset A in E (hence
A is at the same time a K and a G) vhich is not capacitable.
For each. compact KR , denote by A(y)and 8(y)the
respective diameters of the sets K fq Dy and K tidy, where
Dy and dy designate respectively the half-lines (x _ 0, y) and
0, y).
Let ?(u) be a continuous, and increasing real function,
defined for u > 0, and Such that ?(0) l ' and (q-.) 2.
(For example, ?(u) 2. e-).
Set '(y) -- ?(AK (y). oK(y)) and 'f(K) ' fr d/s(y) dy,
()
the integral being taken on the projection P(K) of K on the
y-axis. This integral has a sense, for d/(y) is upper semi-
continuous'. Since l b.2, œ(K) s clearly sub-additive;
it is on the other hand increasing and continuous on the right;
and we can add that /(K)= 0 for each compact K whose
projection on Oy is of linear measure zero.
Now let A be the closed set (x, O; Oy.,___
We have
/,(A) and f*(A) 2.
In fact, '(y)--i for each KA, from which follows
/,(A)-' ! and on the other hand we have [(o)- 2 for each
open set o containing A, for there exist compacts K to such
that (y)> 2-- for each arbitrarily given a > 0.
32. (lapacitability of sets which are not.K-borelian.
section we shall give two examples.
In this
ß 32. I. EXAMPLE. ,, The follo,ing is an example O. f a capa-
city f O, alternating of order ex=, defined on the set X(E) ofall
sub-compacts of a compact space E, for hich there exists a
non-capacitable set A E vhich is at the same time a K f] G
and a G.
Let X be the compact space obtained by adding the point of
Alexandroff to to a discrete space of cardinal number 2a0. Let Y
be the segment [0, ] and let E X X Y. For each compact
K cE, let
/(K) '--the linear measure of the projection of K on Y.
Then [ is indeed a capacity of order e L.
Now by hypothesis there exists a l-i correspondence given
226 C'STXVS COQV
by y '-?(x). from (X e0) o.nto Y. Designate by A the graph
(that the set of points (x, ?(x)) where x a (X -o)).
set is of thelform K fq G; on the other hand, for each > 0, the
set A of points (x, y)such thatly ?(x)!< s and x a (X
is open and so A- A is also a G.
Now each sub-compact of A is discrete, and hence finite,
from which it follows that /(A)0. Each open set
containing A projects onto Y;it follows that f*(A)
Hence, A is not œ-capacitable.
32.2. Exarns. We shall no, present an example of a
capacity f O, alternating of order et= defined on the set Z(E) of
compacts of a locally compact space E and for ,hich there
exists a closed set A E hich is not f-capacitable.
It suffices to modify the preceding example by designating
by X a discrete space of cardinal number 2t0. The space
E X X Y is locally compact and the graph A of 9 is the
required closed set.
32. 3. REMAlK. Th. ese two examples show that the
statements of the preceding theorems cannot be extended,
without some restrictive 'hypothesis on the space E, to every
element of the borelian field generated by the open and closed
sets of E even when we impose on f the greatest regularity
possible; examples of restrictive hypotheses on E which
would be sufficient are the following; E is a complete, sepa-
rable metric space; or E is compact and such that each open
set G of E is a K,. Examples 32. I. and 32. 2. justify the use
of the K,borelian and K-analytic sets,
33. (lapacitability of sets CA.. .... It is' well known that, for
each Radon measure ,, which is defined, for example, on the
plane R , each set CA (that is to say the complement of an
analytic set) is ,-measurable. We cannot
result for capacities however regular they
precisely, we have the following theorem.
state the same
mav be. More
33. I. TSEORE. If E- R-"' and g ;(E), the statement
there exists a capacity [0 of order a on $, and a CA E
hich is not [-capacitable is not in contradiction g,,ith the ordi-
nary axioms of set theory.
THEORY OF CAPACITIES
227
Proof. According to a result of Novikov [i] which appears
to have been previously stated without proof by Goedel, the
statement there exists on the real line R a projective set of
class P. which is not measurable in the sense of Lebesgue is
not in contradiction with the ordinary axioms of the theory
of sets (being admitted that these axioms are consistent).
Now let A be a straight line of E- R and B a subset of A
which is projective of 'class P and is not measurable in the
sense of Lebesgue. For each compact K E, set/(K) equal to
the linear measure of the projection of K on A. It is a
capacity which is 2> 0, and it is of order ct on . (-E!.
There exists(') a subset AE whose projecton on A is
identical to B, and which is of class C,, that is, the complement
of an analytic set.
This set A cannot-be /-capacitable, otherwise the set B
would be measurable in the sense of Lebesgue, according to
Theorem 22.3. of Chapter v.
In what follows We shall make use of the fact that there even
exists (') in R a set CA of interior f-capacity zero and whose
orthogonal projection on A is identical to A:
Indeed, the projective set of Novikov is of class B; that is,
the projective set and its complement are of class P. It
follows easily that thee exists a.part!tion of A into two se. ts
of class P.,. each of which has its ntenor measure zero and ts
exterior measure infinite.
Each of these two sets is the projection of a set, say A
(i ,, l, 9), of Pt a which is of class CA, and we can.always make
'them such that A, and Ao. are contained in two disjoint open
sets. As a result of this precaution and since [, (A,) f, (A.) 0.,
We also have [,(A, U A) -- 0. The set (A, kJ A), Which s
still of class CA, possesses the required property.
33.2. Consequence. It follows immediately that if,
in E R ' for ex.a.mple, a set is measurable for each Positive_
Radon measure, t s not necessarily capacitable for each capa
city which is > 0 and is of order ext.
In the same line, we can set the following problem.
(s, ,) The words there exists are a convenient abbreviation for ½ there is
no contradiction in supposing that there exists ,.
228 sAv.
33. 3. Problem. If A is a subset of the plane E ' R ' (for
example) which is of measure zero for each Radon measure
without point masses, is A capacitable for each capacity
[ : 0 and of order c% on ,i (E)?
34. Construction of non-capacitable sets for each sub-additive
capacity. Let 8 be an additive and hereditary set of com-
pacts of a Hausdorff space E, and let f be a sub-additive capa-
city (hence 0)'on .8. (For example, f is > 0 and of order
ex, with n > 2.
According to Lemma 7.9., we have, for any A, B C E:
34. 1.
f*A U B) < f* (A) + if(B).
Furthermore, let K be such that K(AUB)with K a';. For
each open set o such that B o, we have
K -- (K--o) U (K fi o),
with
(K
Therefore
f(K) f*(K
o) q- f* (K 0
f(K o) q- f*(K f o)<f,(A) q- f*(½o).
We can find a sequence (K. ,,) such that f(K.)-.-/,(AUB)
and/(,)--,-if(B). Passing to the limit, it follows that
34. 2.
f.(A U B) f,(A) + F(B).
We have, of course, an analogous formula by interchanging A
and B.
Then let C be an f-capacitable set with f(C) > 0. If there
exists a partition of C into two sets A, B such that
f.(A) .... f,(B) .. 0, the inequality 34, 2. gives f(C) < if(B); and
since B cC, we have/(B) '-- f(C)> 0.
Similarly
F(A) --- f(C) > O.
The sets A and B are therefore not œ-capacitable.
Suppose now that C is a metrizable compact having the
cardinal 2a0. By using the axiom of choice, we can easily
partition C into two sets A and B such that each subcompact
of C having the cardinal 2a0 intersects 'A and B. In other
words, each subcompact of A or B will be at most denume-
ruble.
TIOaY O CarXCs 229
Now if f is such that f(K) -- 0 for each compact containing
only one Point, the sub-additivity of f implies [ (X).'-' 0 for
each X which is at most denumerable. Then if œ (C) > '0, we
have the following for the sets A and B:
f,(A) f,, (B)-.-- 0
and
f (A)- F(B)-f(C)>0.
They are therefore not capacitable.
35. Intersection of capacitable sets. We have stated pre-
viously, that for the capacities [ of order eta, the intersection
of two /-capacitable sets need not be /-capacitable. The
reason for this is as follows. Let A be a set which is not
œ-capacitable and let B,, B be two disjoint sets such that
(AUB,) and (AUB.) are f-calpacitable; their intersection is
identical to A, which is not f-capacitable.
Here are two examples where this construction is appli-
cable.
35. 1. EXAMPLE. Let f denote the Newtonian capacity
in the spa. ce E ... R a. We shall designate by A a bounded
non-capactable set (there exists such according to section 34)
and by B, B._, two disjoint concentric spheres each of which
contains A. We have /,(A U B) --/*(A U B) --/(B) accor-
.ding to the classical theory of potential; hence, (A tAB,) and
(A U B) furnish the required example.
35. 2. ExAPLS. Let f(K) be defined on the set of
compacts of the plane E R as follows: f(K) linear
measure of the orthogonal projection of 'K on a straight line
A of R . Let A again denote a bounded non-capacitable set
(construct A by the method of section 34 or by using Theo-
rem 33. 1.). This time B, and B are two disjoint concentric
circumferences containing A. It is immediate here that
f,(A U B,) -- f*(A U B,) -- f(B,) (i -- l, 2).
36. Decreasinõ sequences .of capacitable sets. In spite of
the fact that for the capacities of order c%, the intersection of
two capacitable sets is not always capacitable, we could hope
that the intersection A of a decreasing sequence of capacitable
-sets Aa is capacitable and that limf(A)- f(A). Let us show
-that neither of these two results is correct.
230 USTAW. CUOQU.T
Recall, for instance, Example-35.2. The set A will still
denote a bounded non-capacitable set. Let B0 be the cir-
cumference of a circle of radius containing A and let B. .be
the circumference of a circle concentric to B0 and of radius
(t -1- x) where x > 0. Denote by C the open annulus bounded
by B0 and B.
If we set a,- (A U C,/n) we have A (' A, since (' C,/,-- ½.
Now each of the sets A is /-capacitable, the sequence of
A, is decreasing, but their intersection is not f-capacitable.
i )
On the other hand, we have f(C,t) 2 q- - and f()- 0.
Hence it is not true that limf(C,/)- f(' C,/a) although the
C,/ constitute a decreasing sequence of plane open sets, that
is, sets of very regular topological structure.
We could easily construct an analogous example for the
Newtonian capacity f in the space R .
37. Application of the theory of capacitability to the study
of measure. We shall give three examples of the application
of the theory of capacitability to the .study of measure.
37. t. EXAMPLE. Let A be a borelian or analytic set in
the plane E ", R , and let A be a straight line in the plane.
Let us suppose that the projection (praA) of A on A has a non-
zero linear measure. Since A is analytic, it is f-capacitable for
the capacity f defined in Example 35. 2.
Therefore, for each O, A contains a compact K such that
mes.pra A... mes pra K < .
This result can easily be improved in the sense that we can
choose the compact K such that it contains at most one point
on. each straight line perpendicular to A; the projection then
defines a homeomorphism between K and (pra K).
Notice that the same property cannot be demonstrated if we
replace A by a set which is the complement of
set, even if its projection on A is identical to A.
from the second example studied in section 33.
an analytic
This follows
37. 2. EXAMPLE. More generally let A be a. K-analytic
and completely regular space, and let ? be a continuous map
THEOIlY OF CAPACITIES
231.
of A into a locally conapact space F on which there is defined
a positive Radon measure .
There exist some compacts K A such that ((K)) approxi-
mates (? (A))
arbitrarily closely.
The proof is entirely analogous to that of the
example.
preceding
37.3. EXAMPLE.
compact space E, and
of which intersects A.
Let A be a K-analytic subset of a
let 3{ be a set of subcompacts of E each
Let us suppose that , in the topolo-
gical space F of subcompacts of E, is I-measurable for a cer-
tain Radon neasure 0 on this space F, and that (31,) >. 0.
Then for each s > 0 there exists a subcompact K A such that,
if ' denotes the set of elements of ,t which intersect K, we
uses the fundanental scheme of
have (.) ,(') <
The very simple proof
capacities of order
38. The study of monotone capacities of order
We shall not make a direct study of capacities of order A, but
we shall use the properties already established for the capa-
cities of order c%. Thanks to the notion of conjugate
capacities which we introduced at the end of Chapter
'sce 5. 6.), to each of the properties of capacities of order
there corresponds a dual property for capacities of order
This duality gives some substantial results only for capacities
defined on a set of closed subsets of the space E, but this par-
ticular case appears to be sufficient for the study of capacities
of order
38. t. Tnroar. M. Let E be a completely regular Hausdor#
space, .let g be an additive and hereditary class o[ sub-compacts
o[ E, and let [ be a capacity o[ order ,1, ( . t, b) defined on
g with (sup [)< +.
(i) Each A E such that (E A) is K-analytic [or one of the
compact extensions (,,.o) E of E, and such that (A) c g, is [-capa-
citable.
.(s0) It would be interesting to find general cases where this property (that (E . A)
is K-analytic) would be independent of the considered compact extension E. We
find in neider [l], [2], some theorems in this sense, when the considered compact
extensions have a certain character of denumerability.
16
232
GUSTAVE CHOQUET
(ii) If in addition g is identical to the set '(E) of compacts o[
E, g,,e ha,e [.( a) - li,n/,(An) for each decreasing sequence of
subsets An of E {property a,.); and each. denumerable intersec-
tion of [-capacitable sets is [-capacitable.
Proof of (i). Let us designate by f the extension of f to the
set .q.(E) of compacts of E. According to Theorem 16. 3.,
for each A E such that :.(A) (z: g, the /-capacitability of A
is equivalent to its /-capacitability. In order to study the
/-capacitability of such sets A, we can therefore suppose hence-
forth that ; .;(E).
The hypothesis on the class of œ remains the same because
according to Theorem 18. ., since (E) is G-separable and
nultiplicative, the extension of [ is still of class and we
still have (sup [)< .
Let E be a compact extension of E. According to the
remarks at the end of section 20, the extension of œ to the set
ß () is still of order l/b since 2> (, b), and the character
of capacitability of the sets A which were considered remains
unchanged in this new extension; their interior and exterior
capacities also remain unchanged.
We are therefore brought back to the study of the much
simpler case where the space E is compact and where
(We now us__e the notation E in place of E).
Then let [ be the conjugate capacity of [ which was defined
at the end of Chapter nx (see 15. 6.). Thi.s capacity is of
order and is > . Therefore, according to Theorem
30. 1. above, if (E A) is a K-analytic set, (E A) is -capa-
citable, and thus A is/-capacitable.
(ii) .If
ß Proo[ o[ . g .(E), we need only the second exten-
sion used above n order to reduce the proof to the case where
E is compact. Now'if A,, is an arbitrary sequence of subsets of
E, their interior and exterior capacities and those of A- 'q A
remain unchanged in this extension.
We can therefore suppose that E is compact, and the conju-
gate capacity allows us to interpret Corollary 28. 3. and to
obtain the second part of the theorem.
38. 2. CoaonnaaY. If E is homeomorphic to a borelian
(in the classical sense) subset o[ a complete, separable metric space,
THEORY OF CAPACITIES
233
and if f is a capacity of order 11o (,a.. (, b)) hich is defined
on the set .(E)of compacts of E (sup f) ... o, then each
borelian subset of E or each set ,,bose complement 'is K-analytic
s f-capacitable.
Indeed, there exists in this case an extension of E such
that E is compact and metrizable. If A is borelian in E, or
has a complement IA which is analytic, the same holds in
E. Hence, we can apply Theorem 38. l.
38. 3. Raaa. Since the CA sets, whose topological
nature is not well known, are those which are capacitable of
order Ag, it follows that the capacities of order A.,b are in a
certain sense less ½½ natural than capacities of order
Starting with these two classes, one can construct capaci-
ties with curious properties. For instance, if [ is the sum of a
capacity of order et and a capacity of order AI, on the set of
subcompacts of E R , for example, every borelian set A c- E
is f-capacitable, but it is po. ssible to construct f in .such a way
that ( there exist analyt.c sets and sets 'CA which are not
/-capacitable.
38.4.
triction
REMARK.
(sup. f)<
Ee
linear measure
pectively.
The following will show that the res-
is essential in the preceding theorem.
Let E R", and let x'x, y'y be two perpendicular axes in
For each compact K E, let ,(K) and y(K) be the
of the intersection of K with x'x and y'y res-
Set
f(K)--. .:(K). ?v(K ).
Then f is a capacity of order ,ll/b. The continuity on the
right is obvious. Let us now set K K Vl(x'x) and
Ky ' K fq (y'y). The applications K- K:. and K- Ky are
fq-hoxnoxnorphisms and so is the application K- K. X K s.
Now if ¾ denotes the Lebesgue measure in R , we have
f(K) -- 3(K). ?v(K) v(K:, X Ky).
Since is of order =, then f is also.
Now if A is the straight line x'x., .we have/.(A) -- 0.
However, for each open set o cøntamng A, we have f(o) -- q- o,
234 (;USTAVE CHOQUET
and hence f*(A) -- + c. Then A is not capacitable, although
A and its complement are very simple borelian sets.
Observe that if E is a compact space, if 8'--/(E), and
if f(K) % + for each K g, we have (sup. f)___/(E)< + .
38. 5. REMnant. "" For capacities of order eta, we do not
have [, (('] A,,) -' lim [, (A,) for each decreasing sequence of
sets a,. Si,nilarly, we do not have œ* (J a,) .-- [(a,) for every
increasing sequence of A,, when f is of order b,.
The following is an example in which the capacity œ and the
class g are, however, exceptionally regular. The set E is the
segment [0,] and g--(E). We set /(K)--0 except
when K--E (/(E) ). This capacity is of order and
every subset of E is capscitable. However, if A, is a strictly
increasing sequence of compacts of E such that .JA,
have
lim/(A,) 0 and [(tJ A,,) -- .
CHAPTER VII
EXTREMAL
ELEMENTS OF GONVEX GONES AND
REPRESENTATIONS. APPLIGATIONS
INTEGRAL
39. Introduction. We propose to study some convex
cones whose elements are real-valued or vector-valued functions,
to find their extremal elements, and to use these for integral
representation of the elements of these cones.
These representations will furnish in certain cases a simple
geometric interpretation of the elements being studied, and they
will enable us to show their relations with other problems.
Throughout this chapter the vector spaces under conside-
ration are assumed to be spaces over the
fact will not be mentioned again. The
made for all cones.
Let us
real field R, and this
pti '
same assam on IS
first recall a few classical definitions and results
(see also Bourbaki [4]).
39. . Extreme points and extremal elements. . Let g be a
vector space and ½ a convex subset of . We shall say that
a a ( is an extreme point of ½ if no open segment of ½ contains a.
Now let ½ be a convex cone in which contains no straight
line passing through the origin. If Y6 is an affine subspace
of , which does not contain 0 and which meets every ray of ½,
then a a ½fl Y6 is an extreme point of ½ [q Y6 if and only if the
equation
a a, q- a with a, and a a ,
implies
and
where ),, and ), are non-negative.
Such an element a a ½ is called an
extremal element
of the
236 avs,v CHOQUET
cone ½; obviously every )a (), '> 0) is then also an extremal
element of ½.
39. 2. Tl.oa. oF KREIN AND MILMAM. - If the ,ector space
is a locally convex Hausdorf[ space, and ½ a con,ex, compact
subset o[ , then the set e(½) o[ all extreme points o[ ½ has
a con,ex hull vhose closure is ½.
In other words, if x a ½, then there exists for every neigh-
borhood V of x a finite number of positive point masses located
at extreme points of ½ and having their center of gravity in V.
The set e(½) is not necessarily compact, If it is compact,
then the preceding theorem can be sharpened as we shall see.
39. 3. Center of lravity.--- Let ½ be a convex compact
subset of a space f, and , a positive Radon measure on ½.
It is possible to find an ultra-filter, weakly converging to ,
on the set of all elementary positive Radon measures defined
on ½, each of which consists of a finite number of point masses.
The centers of gravity G() of these measures are in the com-
pact set .; hence, they converge with respect to the given
ultra-filter to a point G of ½.
Let us show that G is unique. We have, for every continuous
linear functional l(x) on g,
/(G(,)) f d, .: .f l(x)d,,.
Since l(x) is continuous, we obtain
l(G),f
Now, if /(G')--,)for every l, then G'. G. Thus
well-defined by which is sometimes written as
C 15
G f d,-- f xd.
In particular, let us suppose that is the space of all real-
valued functions x '. x(t) defined on a set E. We shall
topologize f by means of the topology of simple convergence;
that is, the point x--0 is assumed to possess a neighborhood
basis of the form V(s, t,,..., t,) consisting of all points x for
which iX(ti)!< (i---[, 2,..., n). This space f is a locally
convex Hausdorff space.
TtIEORY OF CAPACITIES
237
For every t a E, the function l(x)--x(t) is a linear con-
tinuous function on 2. Hence, with the preceding notations,
and designating by Xo(t) the center of gravity of a measure on
C, ve have
xolt) l',ts. - f x(t)
for every
tE.
39. 4 TIsOaS. If the ,ector space is a locally con½,ex
Hausdorl space, and if ½ is a con½,ex compact subset of ,
then for e½,ery Xo ½ there exists a measure ,o.0 on e(C)
hose center of gra½,ity is Xo.
Proof. For every neighborhood V of x, there is a neasure
, on e(½) of total mass 1, which consists of a finite number
of point masses, and which has its center of gravity-G(,) in V.
Hence, there exists an ultra-filter on the set of these such
that the associated G() converge to x0. But, on the' other
hand, the measures L i converge weakly to a neasure 0 whose
support is e(½). The total mass of ?-0 is 1, and its center of
gravity is indeed x0.
If e(() is closed,.e(½) is obviously the support of .
39. 5. RzAa. It would be interesting to know whether
it is always possible to impose on the measure 0 the condition
that its support be e(½), in other words, that [½..-e(½)] have
0-measure zero.
It should be observed that if is a normed vector space,
then e(½)is a Ga. In the general case, little is known concer-
ning the topological character of e(½).
39. 6. ArrLCATIOrq. Suppose that is the vector space
of all real-valued functions defined on a space E, with the
topology of simple convergence.
Let ½ be a convex cone of f, and assume that there exists a
point to a E such that X(to) . 0 for every x a ½.
We designate by ½ the set of all normalized elements of
½, that is, the set of all x a ½ for which X(to) .... 1. We further
designate by e(½) the set of all extreme points of ½.
If ½, is compact, then the above theorem shows that for
every x a there exists a measure .> 0 on e(½) such that
for every t a E.
238
GUSTAVE CHOQUET
In almost all the
compact set.
c&ses
which we shall study, e(tq,) will be a
39. 7. Uniqueness of the measure associated with an
element x ½. Suppose again that is locally convex, ½ a
convex cone in which contains no straight line passing
through the origin, and gg a closed linear variety in which
does not contain 0 and meets every ray of ½.
Let us assume that ½, . ½ FI is compact and has the property
(even in the case where e(½,) is not compact) that there exists,
for every xa ½,, one and only one measure of total mass
whose support is e(½,) and ;vhose center is x.
Then there exists, for every x a ½, one and only one measure
?. on e(½) for which x, ,fxd?.(e). We shall denote this
integral by x().
This correspondence
between the measures ,..>0 on
and the points of ½ is one-to-one, and since, moreover,
x(, + ), ,x(,) + x(), and x(),,)---),x(,) for ), O,
this correspondence is an isomorphism between the order
structure of the set of the , 0 defined on e(½,) and the order
structure of ½ associated with
½(a - b if b '" a q- c).
Since the ordered set of the ?.>0 is a lattice, the ordered
cone ½ is also a lattice. We can therefore state the following
result:
39. 8. THEOREM. If there exists a unique integral rePre-
sentation of the points of ½ by means of a measure on e(½,),
then the ordered cone ½ is a lattice.
The fact that ½ is a lattice may be interpreted geometrically
as follows: if ½', and ½'[ are the sets obtained from ½, by means
of two positive homotheties, (with arbitrary centers) then the
set ½ fq ½' is either enpty or homothetic to ½, under a positive
homothety.
The necessary condition for uniqueness given in the prece-
ding theorem makes it often possible to determine a priori
cases where uniqueness is lacking. It would be very interes-
ting to know if the above condition is both necessary and suffi-
cient for the existence and umqueness of the integral repre-
sentation.
TtIEORY OF CAPACITIES
239
39. 9. Examples of cones which are lattices. -- (i) Let E be
an ordered set, and 3 the cone of all non-negative increasing real-
valued functions defined on E. For any two elements f and f.
/n : such h and f_
of 3, the set of all t at f,- f .- f has a
smallest element [0 i the sense that [o(X)< œ(x) for every
x E, but in general it is not true that f, [, so that 3 is
not a lattice.
But when E is totally ordered, 3 is a lattice.
(2) E is a Greenian domain of 'R" and ½ is the cone of real
positive functions which are super-harmonic in E. The cone ½
is a lattice. It follows immediately that the sub-cone of ½
which consists of the positive and harmonic functions is also
a lattice. The extrenal elements of ½ are the multiples
,G(P 0, Q) of the Green's function with p61e P0 and certain
limits of kG(P0, Q) obtained by letting P0 tend toward the
frontier of E.
The set of
in general; nevertheless, the
means of extremal elements
Martin []).
normalized extremal elenents is not compact
integral representation by
exists and is unique (see
(3) When n _ in example (2), ( is identical to the set of
positive and concave functions on an interval (a, b) of R. We
night believe, more generally, that if E is a convex set
of R" and ½ is the set of all positive and concave functions
on E, then ( is a lattice. This is not true.
For example, let E be the circle x'" q-y_< ! of R ', and let
i : :: ! x, 6.. TM 1.-4-x. If f ,'-- f . did exist, we should have
'[.____<f--nf. (hnear functions greater than [, and [ on E).
Now we have also [... inf. (elements of ( greater than [, and
f on E.). We would therefore hav [,[ [. But then for
each linear function 1 C such , [ l, since this implies
that ,
,, [ l, e would have l-- [+ g where g (. Since [ is not
hnear, ths equality is impossible.
Then the integral representation for tq is not unique, as
will be verified in the following particular case: It is immediate
that the functions (l---x), (1 d- x), (ly), (. + y) are extre,nal
elements. No,v 2--(lx)+ (1 + x) and 2(t--y)+ (1 + y).
Thi s proves the non-uniqueness of the representation of the
function f-- 2.
240 avsxv cioQv.T
40. Extremal elements of the cone of positive increasin 9 func-
tions. If E and F are two ordered'sets, each homomorphism
of E .into F is called an increasing application of E into F. If
we wish to define .the sum of two such applications and the
product of one of these applications by a real constant, we are
led to suppose that F is a vector space on the field R.
In order to obtain substantial results, we shall suppose
moreover that F is a vector lattice on the field of reals, and that
E is filtering on the right. We then have the following
statement.
40. 1. Ta.oa... Let 3 be the con½,ex cone of the positi,e
and increasing applications f of a set E, hich is ordered and
filtering on the right, into a ½,ector lattice F. The set of extremal
elements of is identical mith the set of elements f of ; hich,
.besides the value O, take at most only one value which is ---% 0 and
ts extremal on F+; any such [unction [ is of the [orm
where A is a subset of E which is hereditary to the le[t ('), and b
is an extremal element on the cone F+ of elements > 0 of F, with
0 for x a A,
b for xa IA.
Proo[. It is immediate that the set of positive and increa-
sing applications [ of E into F is a convex cone. It is likewise
immediate that there is identity between the elements [a ;)
such that/(E) contains besides 0 only one extremal element of
F+ and the set of h, .
Then let f be a function of the form/,(x).
Suppose that [= [, q- f, where [, and f a 3.
For 'each x a A, ,ve have 0- [,(x) q- f(x) and thus
f,(.) .-. f,(a:)-0.
Let u and l (A); for each m. u we have
f, (u) + f, (u) = , - f, (w) + L
and
thus
If,(,,,,) .f,(u)] +
f, (,.,.)] = o.
That is, such that (x'__< x and xa A)
TItEORY OF CAPACITIES
241
It follows that
Now since
than u and v.
f, - f, and f, (u) - f,
E is filtering on the right, there exists a w
It follows therefore that 'I
greater
f, (u) - f, (,) and
l', (,,) - f._,
In other words, f,
with b- b, + b,.
F+, ;ve have b,
real numbers > 0.
and f, on [ A take constant values b, and b.
Moreover, since b is an extremal element of
k,b and b X,b where X, and ), are two
We have therefore f," X,f and f'
vhich shows that œ is extremal.
Conversely, suppose noxv [ is an extremal element of
If œ(E) contains besides 0 only a single element b =yk 0, it is
clear that [ cannot be extremal if b is not extremal on F+. Let
us show that if œ(E) contains at least two elements b and c
different from zero, f is not extre,nal.
We can always suppose b < c; for if œ(u) and œ(v) are two
distinct elements of/(E), there exists +v E such that u and
v.._<C_ and hence f(u) and [(v)/(w). In other words, /(E)
contains two distinct comparable ele,nents, ([(u), f(w))or
(/(), f()).
Then let f, inf (/, b); f,,. s'up (f, b) b.
We have f- -f, q- f by virtue of the identity
inf (y, b) + sup (y, b) -- y q- b.
Now inf (y, b)and sup (y, b)are two
y; therefore [, and [o. belong to 3.
[ is not identically zero. But we
we have f, 0 when. f b.
f-- f, q- f, shows that f is not
Hence
extre,nal.
increasing functions of
Since sup (c, b)- b /: O,
cannot have fs - X4 since
the decomposition
40.2. R.aar. It is sufficient to reverse the order in E
in order to obtain a characterization of the extre,nal elements
of the set of positive and. decreasing applications into F of a set
E which is ordered and filtering on the left.
40.3. REMAnK. If. there exists on E a topology compa-
tible with its structure of ordered set filtering on the right,
and on F a topology compatible with its structure of vector
lattice, we can, instead of studying the cone 3, study the sub-
cone Y of applications f which are also continuous on the
right on E (in an obvious sense). A reasoning formally iden-
tical to the preceding furnishes as extremal elements of Y the
elements [, which are continuous on the right, that is, those
for which A is a subset of E which is open on the right and
hereditary on the left.
40. 4. Inteõral representation of real-valued positive and
increasinõ functions. - We shall now suppose that E possesses
a largest element o, and that F is the real line R. Let 3 be
the set of elenents f of 3 such that f(o) . It is seen
immediately that is, in the vector space of real valued func-
tions on E (with the topology of simple convergence) a
convex and cronpact subset. On the other hand, the subset
e(,') of extrene points of , is identical with the set of extremal
elements f "f,, of such that [()- ; hence e(3,) is
compact. According to the Application in 39. 6., it follows
that [or each [ :, there exists a positive Radon measure on
e(3,) such that
f(x) .... ,,(x) dy. (A).
We can easily extend this result to the case where E does not
possess a largest elelnent, when we limit the study to the ele-
ments œof :) which are bounded on E. It is sufficient to extend
these functions [ to the set E which results from E by adding to
it a largest element o.
/tO. 5. EXAMPLE. If E is the interval [0, ] of the real line,
with the usual order, the extremal elements of 3, are of the
for,n with A -- set of x < a or set of x .< a, (0 a < t).
It is easy to see that e(3,) is homeomorphic 'to the se--of'all A
with the topology of order (the order here being defined by the
relation of inclusion, A, c,A.).
40. 6. Interpretation of the formula f(x) .fL,,(x) d[,..
The class of the f, can be identified with the class of subsets
A of E which are hereditary on the right or with the class of
THEORY OF CAPACITIES
243
their complements A' which are indeed the hereditary on the
right subsets 'of E. Now
l& when x a A',
[" when x a A.
Therefore, f(x) is just the [-measure of the compact set of those
A' which contain x. If the set of A' is ordered by inclusion,
and if we denote by A' (x) the set A' of points of E greater
than x, we can still say that f(x) is the -measure of the set of
an A'
40. 7. Uniqueness of the measure . When E is .the
interval [0, 1] the cone , is a lattice and it is well known
that there is a unique measure , determined by an increasing
fonction [ on E.
When E is the ordered set R it is no longer true. For
instance let f,(, )= o if ;. < 1 and f, t elsewhere; let
L(, n)' 0 if n < and fa "i elsewhere. T.hose functions
are extremal elements of ; however the funchon
has another. representation in ierms of extremal elements:
f--f q-f,. where f -'-0 if ,<< t and - .. 1 and f '" . else-
where; f; 0 if < I or I and f,-- elsewhere
4i. Extremal elements of the cone of positive and increasin 9
valuations on a distributive lattice. Let E be a distributive
lattice and F an ordered vector space. Recall that a valuation [
of E into F is an application of E into F such that
f(a b) q- f(a b) -- f(a) + f(b).
It is clear that these valuations constitute a vector space. We
shall designate by the convex cone of valuations of E into F
;vhich are positi,e and increasing.
41. 1. THEOREM. The set of extremal elements of ' is
identical ,ith the set o[ [unctions o[ the [orm [, x, gehere P denotes
a partition of E into
hereditary on
s an extremal
with
to sub-lattices E, (P) and E,P) geith E, (P)
the le[t, E(P) hereditary on the right, and where
element o[ the cone F+ o[ positi½,e elements of F,
if x L,
if x L,.
244
' GUSTAVE CHOQUET
Proof. (a) We easily demonstrate at once the identity
between the functions œ a ß which take only the values 0 and k,
and the functions of the form [,,. Then, exactly in the same
manner as in the Preceding theorem, we show that each
element f,,. is extremal for .
(b) Conversely, suppose that f is an extremal element of .
If f(E) contains, other than zero, only one element k--=0, it
is clear that f cannot (eEextrema. 1 if f is not extremal on 'F'
Let us show that if ) contains at least two elements
and which are different from zero, then œis not extremal. As
in the proof of Theorem 40. i. we can always suppose X <
Let a and b be two points of E such that f(a) "' ), and f(b) --
Let
f, (x) = f(z,_, a) f(a); f, (a:) f(x,.-. a).
We clearly have f, and f_O and f,, f, ncreasing. On the
other hand, since E is distributive, we verify easily that œ,
and [ are two valuations. We have therefore œ, and [ a
and of, ', -I- f-'
N (a) -- 0 and f, (b) % 0. Since f(a) =/= 0, we cannot
have [, ,.. k,[ where k, is a constant. This fact shows that [is
not extremal.
41.2. Rsaa:.. We can remark, as for the theorem
of 40. l., that if E and F possess topologies compatible with
their structures, the extremal elements of the sub-cone ½' ½
made up of the continuous on the right elements of ½ are
those of.the functions. œ, which are continuous on the right.
In fact, n the prece&ng proof [, and [ are continuous on the
right if f is continuous on the right.
4i. 3. Integral representation and interpretation. When F
is the real line R, we obtain for the integral representation of the
elements of ß results quite analogous to those relative to the
cone , either when E possesses a greatest element or more
generally when the function œ that we wish to represent is
bounded on. E. This integral representation results from the
fact that the set of normalized extremal elements œ,, of is
compact for the topology of simple convergence. The formula
f(x) ...,.. f fj,, , (x) d?. (P),
THEORY OF CAPACITIES
245
valid for ea.c.h x, shows that [(x) is equal to the ?.-measure of the
set of partmons P such that x L(P).
It would be interesting to know whether ß is a lattice and
whether the integral representation of the elements of q is
unique.
42. Application to the inte{lral representation of simply additive
measures. Let E be an algebra of subsets of a ven set A;
that is, if X, aaE and X. a E, we h. ave X, U X.., a E and l (X,)a E.
The set E is distributive lattice when E is ordered by the
inclusion relation on A.
Let A'I,, be the set of positive and simply additive measures
on E. Each of these measures is clearly a positive and in-
creasing valuation on E. Conversely, each positive valuation f
on E is of the form [(9)q- (a measure); in fact, [[--/()] -- g is
a positive 'valuation on E with [(.e/) 0. Now
g(X U Y) q- g(½) -- g(X)q- g(Y)
when
XfY--;
it follows that.
g<X O Y) -- g(X) q- g(Y).
The extremal
rive factor, the
and .
elements of .t10 are, therefore, within a posi-
measures œ on E.. which take only the values 0
The set .r10, of measures [ on E such that/(A)- 1 and its
ß subset e(l'ba,) of measures with values 0 or I are compact for
the topology of simple convergence.
Therefore, to each /aA'b, is associated a Radon mea-
s'ure Iz > 0 on e(l/,',a,) such that
f(x) for each X a E.
Let us study the extremal elements of ,lb,,. For each
/ae(A/,,) let B(/)be the set of elements of E such that
/(X) 1. This set constitutes a base of a filter on A such
that if X a B(/), then whenever two elements X,, X.: of E
form a partitio.n of .X, one of them belongs to B(f).
Conversely, ,f B ,s a base of a filter on A, made up of. ele-
ments of E and possessing this last property, we say that B is
246
GUSTAVE CHOQUET
saturated relative to E.
fs on E by placing
To each B ;ve can
associate a function
f(X) ,
when X belongs to the filter of base B,
otherwise.
It is seen immediately that and that B(fr) B.
We ha½,e then established a canonical and one-to-one correspon-
dence between the normalized extremal neas.res on E and the
bases of filters on A (s, hicb are saturated relati½,e to E.
42.1. RsAa. The property which defines the satu-
ration of B resembles the property which characterizes the
ultra-filters, and it is actually identical to it when E % 2 ^.
However, if E -2 a, there exists some bases of filter B satu-
rated relative to E and which are not bases of ultra-filters.
We are now going to make a more detailed study of ,:14,,.
when E ; 2 .
42. 2. Extremal elements of the cone of positive measures on
E'-2 . By using a method of Stone [[], [2], [3] with
a slightly different language, we are going to show how one
can interpret the extremal elements of b, and represent each
element f of
The following could be extended to the case where E is an
arbitrary algebra of subsets of A, but with a more complicated
formulation.
(!) Extremal elements of
satured relative to E are identical with the
Therefore, the extremal elements of
f,,(X) where u is an ultra-filter on A, with
The bases of filters on A
ultra-filters on A.
are the functions
ii when Xau
/(X) "0 when X u]
For example, for each Xo a E, the ultra-filter Uo of the sets
containing X,, corresponds to the point measure fo.--so.
(2) Topology on the space U of ultra-filters. By defini-
tion this topology would be the topology of simple conver-
gence on the set of associated measures [, The space U is
therefore compact. For each X A, let o(X) be the set of
ultra-filters on X. It is immediately seen that for each u0 U
THEORY OF CAPACITIES
247
the set of oX), where X a u0, constitutes a base of neigh-
borhoods of u0.
In particular, for each x0 a A the ultra-filter %0 possesses
a base of neighborhoods formed by u ø itself. Hence u.,, is
isolated in U; conversely, each isolated element of U is of this
For each X A, the set of ultra-filters on X is compact;
hence, o(X) is compact and so is o(A--X). Now each ultra-
filter on A is supported by either X or (A X); therefore
o(X) and co(A- X) constitute a partmon of (Y. Hence, the
sets co(X) are both open and closed.
In particular, each point of U possesses a base of neigh-
borhoods of the form o(X), and hence both open and closed.
It follows conversely that .each subset of U which is both open
and closed can be written uniquely in the form o(X).
For each open set U, let I be the set of isolated points
of 2. Since each point of U is the limit of isolated points,
we have I . Now let X, be the canonical image of I in A.
We have - I o(X,). Therefore, the closure of each open
set is an open set.
The points of U represent the ultra-filters on A. Let us
see how the filters on A are represented in U. Let ; be a
filter on E; there corresponds to it the filtering decreasing set
of open and closed sets (X) where X a . Set ? =- o(X).
x½
We therefore have associated to the closed, non-empty
of U. Conversely let ? be any closed, non-empty set of U.
The set of its open and closed neighborhoods has a canonical
ima. ge in E which is clearly a filter which we denote by ,{?).
It s seen immediately that J((0))-0 for each filter
on A. We ha,e therefore established a canonical and one-to-
one correspondence betveen the filters on E and the closed sets
of U.
In this correspondence, the intersection of a family of .filters
on A corresponds to the closure of the union of the corres-
ponding closed sets in U; the upper bound of a fanily of filters
on A corresponds to the intersection (assumed to be non-
empty) of the corresponding closed sets in U.
248 GUSTAVE CHOQUET.
We shall see later, in the study of capacities of order
which topology it is natural to define on the set of filters on E.
42. 3. Intetral representation of a measure on E 2 a.
According to the general theorem 39. 4., there exists for each
measure f a AI,,, a positive Radon measure I on U such that
f(x) _, ff,, (x)a
for each
XE.
In other
f(X) ' -measure
f(X) is the measure
the supersets. of X.
words, for each X we have
of the set of u supported by X; that is,
?.[to(X)] of the image in U of the filter of
This Radon measure on U is ;vel]. defined for each open and
closed set; since these sets constitute a base of open sets in U,
this measure ,s. s untque.
Conversely, to each Radon
measure f, which is simply
f(X)-
Each measure
on A by-setting
measure . on U is associated a
additive on E, by the relation
f on E can be extended to the set of filters
-., inf f(x) - inf ?.[ico(x)] = ?.[9(5)]
for each filter 5 on A. Then f() is just the -measure of
its image ?(,) in U.
In this interpretation the fact that an additive measure f on
E is not completely additive
(D . . . (On . . . IS a
subsets of U which
follows from the fact that when
sequence of not empty open and closed
are mutually disjoint, we always have
U to,,__ U o,, and hence, in general, different ?.-measures for
these two sets.
We have defined on the space of
.logy of simple convergence on E
are n one-to-one correspondance
measures on U; hence, there exists a topology on the set of
these measures ?.. As the images to(X) of the elements of E
measures f on E the topo-
2 . Now these measures
with the positive Radon
constitute a base of open and closed sets of U, this topology
on the space of Radon measures on U is identical with the
classical topology of vague convergence (see Bourbaki [3]).
THEORY OF CAPACITIES
249
43. Extremal elements of the cone of positive functions alter-
nating of order on an ordered semi-group. ,, We have
already emphasized the analogy between the capacities 'of
order cx_ and the functions of a real variable which are
completely monotone. We are going to see that this analogy
©
s not only formal, but also that these two types of functions
belong to the same very general class of functions in which
exponentials and additive functions play an essential role.
43. I. DEFINITIONS. Let E be an ordered commutative
semi-group with a zero, all of whose other elements are greater
than zero. Let F be an ordered vector space, and let ex be the
convex cone of the functions which are defined on E and take
values' in F., and which are alternating of Order c (see 3. 1.,
Chapter iII).
We shall suppose F such that each X c- F. which is bounded
from above and which is filtering on the right (-") has an upper
bound.
43.2. DEFINITION. Any application f of E. into F such
that f(a-r' b) f(a) -3- f(b) will be called linear.
It is obvious that any linear and positive f belongs to ex; the
set of those functions is a sub-convex cone of , which we will
exponential when ,4 is a
denote by !,.
43.3. DEFINITION. We say that a function b on E is an
real-valued function such that
0 <'.< i and
To each real, linear, and positive f on E corresponds
nential , -- e -y and, conversely,
does not assume zero values on
linear function f Log l/d/.
the expo-
to each exponential , which
E corresponds the positive
43.4. THEOREM. --- In order that an element f of the cone ct
be extremal, it is necessary and sucient that it be of one of the
two lollowing [orms.
() f is an extremal element of the cone ' of linear elements
(am) We say that X t is filtering on the right if for every a, be X, there is an ele-
ment c a X such that a, b __<_ c.
250 GUSTAVE CHOqUET
(2) f .. (i--.4,)¾, ,here 4 is an eocponential
is an extremal element of the con,ex cone
on E and V
Proo[. Let f be an extremal element of a.
(i) If/(E) contains at most two elements 0 and V---/= 0, we
can set [-n(d 1 -)V., ,vhere d/is a function which takes only the
values 0 I and s such that (,) > 0 for each n.
If ,(a) -= 0, it follows from /,(a; b) > 0 that ,(a 'r b'!-= 0
for each b.
If d/(a)"'(b)--, it follows from (0; a, bSO that
,(a'r b)-- I. We have therefore
and hence is an exponential.
In order that such an f be extremal, it is clearly necessary
that V be extremal on F.
(2) If f(E) contains at least two 'distinct elements ),, , such
that ),, =# 0, we can clearly suppose that they are comparable
in F. If f(a)- X and f(b)- ,, with ), < ,% set
and
-=
f(a),
f,(a:)-= f (a) q- 7,(x ; a)y ,, f (a) +
.. f (a: -r a).
Then, œ-- f, q-/',,.
The function f, belongs to c since the operation x- (a;-I-a)
is a homomorphism of E into itself and since [(as-ra) .[(a) O.
Also f belongs to a since on the one hand (k-)y: -- (. )y < 0
and on the other hand f 0 since the inequalities (k0y< 0
and f 0 imply f(a) + f(x)- -f(x-r a) ?_ O. Now
f ,( b) = f (a -r b)
f(a)>X?. > 0.;
hence/,.is not identically zero.
If f s extremal, we have(ti s ... k[. whe. re
number such that 0 < ),< 1; fact lmphes i is a. r,eal
pamcmar
that f(0)= 0 since f,(0)= 0). We can then write
f(a: "r ,)
..
Two cases are then possible. .
(a) First assume that f is not bounded above on S, that s,
assume that there exists no ,) a G such that [(x) < ¾ for each x.
TItEORY OF CAPACITIES
25t
Now f(x) =< f(x 'r a), and hence f(x) . f(a)_ X.f(x).
cannot have ),, 5/: l, otherwise
f(X)
f(a)
We
and f wodd be bounded.
Therefore X,, .. t, and hence
f(x 'r )
f(x) q- f(a).
This equality is true for any x. It is true also for any a; indeed,
the proof above shows that it is true for each a such that [(a) > 0
since, œ being not bounded, œ takes values > X; on the other
hand if a is such that [(a)- 0 the relations
give
and so
q7,(x; a) __< O,
,.,(0; a, x).<O,
f(O) .- 0
f(x)f(x-ra)
and
f (x) f (x -r a)
f(x-r a)' f(x)= f(x) + f(a).
Then f is linear. If f is extremal on a, f is a fortiori extremal
on
(b) Now suppose :hat [is bounded above on E. Let V be its
upper bound (-% 0), which exists since the set/(E) is filtering on
the right in G. Set g--V [. We have the identity
or
g() + Lg(x)= g(x a) +
By the definition of V, we have
inf g(x) .... 0
and
inf g(x 'r a) -- 0.
By taking the lower bound of the two sides of (l) we obtain
(2) g() : x,,v.
This relation is valid for each a such that 0 < f(a)< V.
[(a)- O, we have g(a) V;if [(a) - V, we have g(a)- 0.
can therefore set [(x)..--- 9(x)
such that 0 ._< ? ..< . Also
g() - ,. v.
The relations (t) and (2) can be written now as
If
We
ß V, where ? is a real function
let us set ½ I -t, and so
and ,(a) == ),;
252
GUSTAVE CHOQUET
hence
(3)
'4, (x = + q,
This relation is-'valid for each a such that 0 . f(a)<C V. If
[(a)- O, ,ve have as in the preceding case [(X-r a) '/(x);
thus, +(x-r a) ,J/(x) and, since qb(a) 1, the identity (3) is
again satisfied.
f f() - v, , dso have f(a -r x) -- V. Therefore
.-._ -- o,
and the identity (3) is again
[" (1--+)V, where ,J/is an exponential.
satisfied.
In other words,
Since fis supposed to be extrenal, it is clear that
also an extremal element of the cone G+.
V must be
Corvr. asr.. -- We no;v have to show that the functions f on E
that ;ve have just studied are indeed the extrenal elements
of .
(1) If [is an extremal element of , and if f -.. [, q- f with
[,, f,., e cx, the f, and f,., necessarily belong to .
In fact, f(0) 0 and (q7/'_).0, which implies
and (q7.)f,'-0 (i .-i, 2).
Thus the relation ,(0;a, b)y .--.. 0 is
that
f i(a -- b) f ,(a) 4- f ,(b)
fi(O) -0
Therefore, since f is extremal on , f, and f,_, are. proportional
to [; thus f is extremal on
(2) Let f= ( ')V where qb is an exponential on S and
where V a G+. We first have to show that we have [
Now f> 0 since 0 + <: i; in order to show that (,)y:< 0,
it is sufficient to establish the equivalent relation (x(7)$l 0.
Now q7/(x; a)q,-- d/(x)--d/(x-ra]-- '(x)(i -(a)) and more
generally,
a(x; lai})- '(x)H{_ .. ,(ai)) __0.
Finally, it remains to show that f is also an extremal element
of . We shall be able to do so only after having introduced
a suitable topology on (see section 44 below).
43. 5 Exar.s' E = R" that is the set of elements of
'the group R" with positive coordinates. Each positive linear
THEORY OF CAPACITIES 253
function f on E is of the form >.]azx (a.> 0). The extrenal
elements of (2 are then the positive multiples of the n functions
f,-- xio
Every exponential [ which does not vanish on E is, accord-
ing to a previous remark, of the form e -a' where a.x denotes
the scalar product of the elements a and x of R.
More generally, each exponential '+((x))can be written
+((x,)) II+(x) where i denotes the restriction of ' to the
xcaxi8.
The restriction , is again an exponential function of a real
variable. Now if (a) % 0 for a > 0, then +(na') 0 for every
n; thus +, cannot take the value zero. If ½,(a)- 0 for all a> 0,
then +(0) I or 0; it is easily seen that conversely, each
of these functions is an exponential.
Thus. each. exponential-hb(x) on
+(x)--' II +(x) where
S can be written as
+i(Xi) e
where
-aiX'(ai,,> O)
or
+i(Xi) .-.- e-OO x
1 if X i --- 0
:t0 if X> 0.
or
43.6. EXAMPLE. E is idempotent, that is, x-r x
every x E.
Each linear function on E is identically zero since
x for
f½) f(x-,. x) + f(x) implies f(x) - o.
If is an exponential on E, then
hence +(x) -- 0 or 1. The set of elements x of E, for which
+(x)- 1 is a sub-semigroup of E, hereditary on the left (that
is, x' % x and x a implies x' a ). For if
then qb (a- r b) I. If +(b) I and a < b. then, since
a 'r b b, '(b) -- 'b (a). +(b) and hence qb (a)
Conversely, for every sub-semigroup , of E which is
hereditary on the left, let)+(x)l if x a, and b(x)- 0 if
x**. Then if +(a)- +(b , is follows immediately that
,(a- r b) -- ,(a).,(b).
254
GUSTAVE CHOQUET
If (a)-- 0 and (b)- , then a > b and therefore
aq.b--a;
thus
+ (a q- b) -- q(a) .-- q(a).
lf(,a)--(b)--0, then d/ (a.r b) -- O , since a and b----t.rb;
hence again, q(a. r b) -- q(a) q(b).
There is thus a one-to-one canonical correspondence between
the exponentials on E and the sub-semigroups of E which
are hereditary on the left.
The extremal elements of et are the functions f, v(x) on
E defined by
0
f. v(X)-, iV
where z is a sub-semigroup of E which is hereditary on the left
and V is an extremal element of
43. 7. Exarrs. E is an additive class of sets. Let Ebe
an additive class of subsets of a set A, the operation 'r being
union and the order on E being inclusion.
To every exponential , on E there is a canonically associated
sub-semigroup .z which is hereditary on the left. Let :*
be the set of complements I X of elements X of z; except for
the case where A a E and where l, * is a base of a
filter.
Conversely, to each filter on A having a base consisting
of elements of E*, there is associated the exponential on. E
defined by/(X)' I if C X a and/(X)- 0 if C X,,.
Th. us ex.ponentials, filters and extremal elements of the cone et
are n ths example three aspects o[ the same mathematical
obiect.
The preceding interpretation of exponentials n terms of
filters now permits a better study of the normalized extremal
elemen. ts f of ct when.e.ver F is the additive group R, and an
extension of the definmon of [ to the set of filters on A.
For such an element f
associated with . Then
(
d/), let T be the filter on A
f(x)_ 01 !f for some T, X XY
f for every a fqY - ]
wgaOlY o c,,cmas 255
More generally, let T and T be two filters on A, and let
f(T, T,) -_ t ø if T, T,
I if T, T,
does not exist,
exists;
that is, f(T,, T,) equals
0
if there exists two elements of T and T, which are disjoint,
otherwise.
For each fixed T it is easily seen that the function
f,(T)- f(T,, T)is an alternating function of order on the
semigroup of the filters T .on A, with the operation w deno-
ting the intersection, that is, T 'r T' denoting the filter each
of whose elements is the union of an element of T and an
element of T'.
When T denotes'the filter of supersets of a set X A, the
func. tion fT,(T) is identical with the function f,(X)considered
earher. The function fT,(T)- f(T, T) is called the elemen-
tary [unction alternating of order v and of index T,.
43.8. Special case. ,,, If E is the set of compact subsets of a
Hausdorff space A, and if/. is continuous on the right, the
'filter T associated with [ is just the filter of neighborhoods of
a closed subset of A. This was shown above (see section
26. i4., Chap. v).
44. Topology of simple convergence on c. Application.: Let
us come back to the general case assuming simply that F is
identical with R, and introduce on et the topology of simple
convergence on E.
The set of. exponentials on E is clear!y compact in the
'topology of snnple convergence; the same s true of the set of
elements of a of the form (! ), where , is an exponential.
We shall now show, by using this compactness and the rate
of the decrease of the exponentials d/on E, that each element
(i-- ) is extremal on a.
We use the fact, which is easy to show, tha if ½ denotes
a convex and compact subset of a locally convex Hausdorff
linear space, for each non-extreme point rn a ½, there is a mea-
sure , > 0 of total ,nass ! which is supported by [e(½) l ml]
2'56
Gus'rAVE CHOQUE'r
and whose center of gravity is m [where e(½) denotes the set. of
extreme points of ½].
(1) We suppose first that E has a largest element o. Each
[ a cx is then bounded and the set et, of the elements of such
that [(to) is compact. The set of extreme points of et, is
identical with the set of extremal elements œ of et such that
f(to)--. Now if is an exponential not- on E, then
inf '- 0 (since d/(na)- ((a)"); hence, sup
Then the set e(et,) of extrene points of ct, is contained in the
compact set &;, of the elements (1 +) where _=/_--. We
shall show that e (ex,) .-
Otherwise, suppose [ ( .... ,) is an element of [80,... e(a,)].
There is a measure on the co,npact set (/;,--Ill) such that
f d.-- !
and
hence
For every a a E and .for
which ,(a) q(a)q- is of
its measure. Then
every s > 0, the
-measure zero.
closed set of t for
For let (s) be
(+(a))" ', (ha)
.f d/,(na)
hence ,u.()
"[,t(a) q- '
>: +
a quantity which tends to 0 as n
Then ,,(a) +(a) for almost all t
Frmn the relation
f -- o,
since
(+(,,)-- +,(,,)),> o
almost everywhere, it follows that (a)- +,(a)for almost
every t.
By passing to the limit, this equality holds at each .point of
the compact support of . In other words, { is identical w. ith
eac. h for which t belongs to the support of ,u.. Then s a
pmnt mass supported by the representative point of (1 ..-+),
contrary to hypothesis.
(2) If. E does not possess a .la. rg. est elenent, denote by
the semi-group obtained by adjmmng to E an eltoncut o, by
THEORY OF CAPACITIES
257
definition greater than each element of E and such that
a -r o . o for every a a E.
Let ½x be the set of applications which are alternating of
order e of cx in F.; then obviously, in order that f it is
necessary and sutlicient that the restriction of f to E be an
element of a such.that sup [(x)< [(o).
According to the preceding, the extremal elements of t
are, within a factor, just the functions ' (l ,/), where , is an
exponential on E. Now, if ½ is an exponential on E such that
, 1, then inf½(x) --0. Also, if + is the extension of ,
cE
to E obtained by setting , (o) 0, then , is an exponential on
E. Then (l ,b) is extremal on c;. This implies that (l )
is extremal on cx; otherwise (l ')- f, q- f,. with f, and cx
and f, fa not proportional to (l ,).
We havesup(i')-- i . supf, q- supf (on E). Theniff
denotes the extension of [ to E obtained by setting [(o) .... sup [
on S, we have (. '-- q-f with , and a and f,,
not proportional to --).
Thus the theorem is proved when F = R.
We now suppose F to be arbitrary. Let us prove that each
f-- (! (9 exponential, V extremal on F+) is an extremal
element of .
Assume that [: f, q- f. ([,, [. a).
For any x a E, if [(x)-- O, then [, (x)-t- [(x)- 0; hence
f, (x) 0.
Iff(a:)O, (l. V.)Vis extremal on F+, so that /, (x) 7d f, //
are cohnear to other words, ;ve may set ,"' ,V
and [-= ?V, where ?, and ? are two real, positive functions;
it follows immediately that ,, and ?. are alternating of order
infinity. But we know that the relation
implies that , and .,. are proportional to (l -); hence f
is indeed extremal on
44. I. Rzaa... When Eisidempotent, each exponential
takes only the values 0 and , so that the elements (---,) of t
are increasing functions on E (ordered by the convention
that a < b if b .--a q-c) which take only the values 0 and i.
258 GUSTAVE CHOQUET
Since E thus ordered is filtering on the right, 'these functions
are extrenal on the cone of real, positive, increasing functions
on E; they are, a fortiori, extremal on the cone a.
Thus the proof that the functions (1-') are extremal on a
is very simple in this particular case.
45. Inteõral representation of the elements of . Let us
suppose at first that E has a greatest element ½o. We suppose
here that F .. R. With the notations used above, for each
Idex there exists a measure 0 on the compact set of
extremal elements (l ,)of a (with + g--i)such that
d/,(x)) d?.(t)
for every x a E.
When E does not possess a greatest .elenent, there still is such
a representation whenever the given function [is bounded on E,
being considered as the restriction to E of a function defined on
E (E VI o).
We shall not consider in the general case, the question of
uniqueness of the measure , associated with the given f.
45. t. The case E: 2 a. We shall assume that E is the
additive set of subsets of a set A, the order on E being inclusion;
assume also that F .: R.
The normalized extremal elements of c; are the eleinentary
alternating functions/(X) associated with some filter T on A.
We shall use the space U of the ultra-filters on A which has
already been introduced.
With each filter T on E there is associated in U a closed
set that will be denoted by o(T)'or simply by T. Thus
o if
fT(X/-- i I_ if
(T)flto (X)--
(T)fqto (X)
With each element [ of et there is associated the capacity ? of
order et defined on the set of open and closed subsets of U by
the relation ?(o(X))--œ(X). This capacity ? can be extended
to the set of all the closed sets of U by setting
?(w)
for each such closed W.
inf (to(X))
wcto(x)
THEORY OF CAPACITIES
259
This extension is equivalent to extending the function f to
the ordered semi-group of all filters on E.
Conversely, each capacity ?0 of order aon .(U)
is characterized by its restriction ø-the set of open and closed
subsets o(X); in other words, there corresponds to ? an element
for a.
Summarizing, we .have established a canonical one-to-one
correspondence between the capacities 0 and of order et on
:.(U) and the functions [ 0 and alternating of order on 2 a.
The topology of simple convergence on the set of elementary
functions fr(X) is identical with the classical topology on the
space of closed sets T of U. This follows simply from the fact
that such a closed set has a base of neighborhoods consisting of
the sets o(X).
To each element [ of et there corresponds a Radon measure
, .> 0 on the space .(U) such that
f(x) ....
for every
Xc A,
and, more generally, for each filter on A.
The uniqueness of will be proved later on when we study
capacities of order et on an arbitrary locally compact space.
Let us add that the topology of simple convergence on a is
identical with the vague topology (which we shall define also)
on the set of capacities ? associated with the elements [ of a.
45. 2. The case E .. R+. Let ct be the cone of real
f'unctions :> 0 and alternating of order c on RS. For a given
[a a, if/((t))- 0 (where (l) denotes the point each of whose
,coordinates is i) then f0 since each f is decreasing and
concave on R a
Thus for each [_=_=_0, there is a ), > 0 such that ,/((t))--i.
In other words, the closed hyperplane f((l))_ 1 of the vector
space of real functions on R+ intersects each ray of a at one and
only one point. Let cx, be the set of elements [ of a such that
f((i)) .= i.
Since F is increasing and concave on RS, each f a ex, has the
property that f(x) < sup [1, x,]. Then a, is compact in the
topology of simple convergence.
260 usxvs co
Now the extremal elements of et, are he
and the normalized functions
n functions f
where the t are > 0 or q- , and Et 0.
It follows that each f a't ,hich is continuous
an integral representation of the f'orm
: x i
]. , ,.--t.x
f(x) -- a.x q- ), q- t' : i:i d(t),
on R.a. has
where is a positive measure
of non-zero vectors t of R".
of finite total
on the set
mass,
The functions f a which are not continuous have formally
the same representation but with suitable definitions to take
care of vectors t with infinite coordinates.
When n--, this expression can be simplified and can be
written, for every f aet, as
f(x) ,, x q-f
e-t
d(t),
where is supported by the compact [0, q- ], with the con.-
vention that
x when t "' O.
46. Extremal elements of the cone of monotone functions of
order o on an ordered semi-group. , Let E and F once more b e
a semi-group and an ordered vector space respectively, . which
have the same properties as in section 43. Denote by ,,ll, .the
convex cone of functions from E to F+ which are monotone of
order , (that is, the k-,, are >0).
For everyfa ,1o, we have ,(0; a)0; hence, (/(0)- f(a))'>O.
Then the function g(x)--/(0) ..... [(x) is 2>0, and (,)g<0.
Thus, g is a bounded element of a.
ConVersely, for each bounded element g of c, if g(c) denotes
its upper bound, the function [ g(oe) g(x) is an element
of 1.
It follows easily that the extremal elements of ,lb are the
THEOHY OF CAPACITIES 26!
functions f-+.V, where is an exponential on E and V an
extremal element of F+.
46. I. EXAMPLe.. Le.t E 2 a be the set of subsets of a
set A, the order on E being the inverse of that defined by
inclusion, and the operation in E being intersection. Let F
be identical with R.
The extremal elements of A}, are identical with the functions
f(X) associated with some filter T on E, where
f(x)
0 if X,T
t if XaT.
46. 2. EXAMPLr. Let E be the semi-group R. and F- R.
The non-zero extremal elements are the exponentials e -
where 0 .< t.< .
46. 3. APPLICATION. The introduction of the topology of
simple convergence on i leads to applications analogous to
those obtained by considering the cone Pt. For example
every continuous function f(x) of the real variable x_..'> 0 such
that ( l)'f"'(). o, for all x > 0, that is, every completely
monotone function of , has a representation of the form
f(,x).,
f
where , is defined on [0, [ and has a finite total mass.
result is the classical Bernstein theorem.
.There exists obviously an analogous representation
continuous completely monotone functions on R"+:
f (r)' ,
This
for
where a is positive measure, with a finite total mass, on. R' L.
Similarly, we could state integral representations for positive
completely monotone functions defined on the open positive
half-line x > 0, or, more generally, on the interior of R. But
these generalizations are merely special cases of a more general
result concerning functions defined on an arbitrary semi-group,
which we shall now study briefly.
47. Alternating or monotone functions of order on an arbi-
trary commutative semi-group. Let E be any commutative
262 ausAv coQu
semi-group, and let F be a vector space satisfying the same
conditions as above. A function œ from E to F+ is alterna-
ting (respectively, monotone) of order oc if all its differences
,(x; l al) are < 0 (respectively, .> 0) for any x and a in E.
The convex cone of these functions is again denoted by t
(respectively Ab).
If a and b a E, we shall write a - b if a- b or if b- a-r c.
This relation is reflexive and transitive.
If a b and b- a, we shall write a - b; this relation is an
equivalence relation compati,ble with the relation -.
Moreover, ff a b and a b' then (a-r a')-- (a'-r b').
Then the quotient set E]p is an ordered semi-group in which
the relation x q y is equivalent to x .. y or y :":' x q- z; that.
is, it is an ordered semi-group which we shall call regular.
In E, if a- b and if f a a, then f(a)<f(b); thus if a b,
f(a) .-= f(b).
Then with the function f on E there is canonically associated
on E/p a function alternating of order oc. We obtain an
analogous reduction when [ a orb. Then in studying a and
it may always be supposed that E is a regular ordered semi-
group; this assumption will be made henceforth.
If E possesses a neutral element 0, we have 0 q 0 q-x or
0- x for every x. Then this case has been studied in the
preceding.
If not, we may embed E in the semi-group E obtained by
the addition of a neutral element 0 to E such that 0 - x for
all x; the study of the elements [ of et and ut'b associated with
E is then equivalent to the study of functions defined on the
set of non-zero elements of a regular ordered semi-group with
a zero. This remark simplifies sometimes the study of c and
47. t. DrrllITIOrqS. () An element a of a regular semi-
group E ithout 0 is called extremal if the equality a b q- c
is impossible.
(2) For every a .,hica is extremal, the function % defined by
%(x)- 0 if x a and %(a)=-! is called the singular [unction
ith the pole a.
(3) An exponential on E is again a [unction '+(x) such that
o + ..,- _
THEORY OF CAPACITIES
263
47. 2. Tauoaa. -Let E be a regular ordered semi-group.
In order that an element f of .ib be extremal, it is necessary and
sufficient that it be of one of the' follo,,ing forms:
f" ?V or
,here V is an extremal element
and , an exponential.
f-
of F._, ? a Singular function,
Proof. When E has a zero, no element of E is extremal and
the theorem is a consequence of section 46. We assume,
therefore, that E has no zero and suppose that œis extremal. Fo r
every a E,
f ,,' f, q- f,_,, where f,(x)- f(x -r a)
and '
f (X) -- f(x), f(x 'r a) -- W,(x;
The functions f, and f. belong to vb so that
(l' f, (x) -- f(x-r a) -' where 0 X < t.
CAsa I. If there is an a aE such that f(x 'r a) /_-- 0, then
L--/= 0..
Now X,œ(x) kœ(a); hence, [(x)/X, [(a)/X,, ..--some V 5/= 0.
Then (1) can be written as V.k_-- X.V.X; hence,
k_ L,X for every a such that [(a 'r x)_--/= O.
But. if a such that [(a..rx)0, the identity (t) shows
that Xa 0 and also X- 0. Then again
Thus [ d/V, where ,} is an exponential. If V is extremal,
then V is obviously extremal on F+.
Conversely, if + is an exponential on E, it is easily verified
that f- V belongs to [b for every V F+.
When, moreover, V is extremal on F+, it can be shown' as
before, by intr. oducing the topology of simple convergence, that
each f 'V s an extremal element of
Casz 2. If a)--0 for every a, f(x) is zero at every
non-extremal point of E.
Now every function f from E to F. which is 0 at every
non extremal point of E is an element of Ab. For in Vy[x- ,. \
all the terms are zero except possibly the first, f(x), whlehs0.
Then. in order for such an f t.o be extremal, it is necessary'and
sufficient that the set of points x where f(x):0 cannot be
18
264 GUSTAYE CHOUET
partitioned; in other words, that this set consists of a single
extremal point of E and that the value of [ at this point be an
extremal element of F+. Then f--,V where ? is a singular
function and V an extremal element of F+.
47. 3. REMARK. There is an analogous theorem
ning the extremal elements of .
concer-
47. 4. EXAMPLE. Let E be the additive semi-group of
real numbers > a > 0.
The extremal points of E are the points x of the interval [a, 2a[.
It is immediate that an exponential , on E other than
is not zero at any point of E; then Log / is a positive
linear function on E. Now it is elementary to prove that such
a function has the form tx. 'Thus, each exponential not 0
on E is of the form e -t
If we remark, on the ther hand, that f(x -4- a)< f(a), then
we deduce that f is bounded on [2a, o[.
We can then prove easily that for every f a c there exists
a measure ?. on [0, oe[ such that and a
function s (x) ;> 0 defined for x > a, with s (x) -- 0 for x > 2a
such that
f (x) '... e - d(t) q- s(x) for every x ;> a.
This result is rather 'remarkable since it implies that [ is
analytic on [2a, !... m[ although the conditions q7,2 0 imposed
on œ have. no local interpretation (since the parameters a
appeanng n , are all > a).
An analogous study of the semi-group E of real x > o
(which contains no extremal point) would lead to the classical
representation of positive and completely monotone functions
on ]0,
48. Vague topology on the cone of increasing functions.
Let E be a locall/compact space and 3 the convex cone of real,
non-negative, and increasing functions f defined on the class
.(E) of compact subsets of E. We have already introduced
on 3 the topology of simple convergence. However, this
topology is not satisfactory for investigation of the subcone
THEORY OF CAPACITIES 265
of : consisting of positive capacities f (that is, the elements f
of : which are continuous on the right).
We shall therefore introduce a weaker topology by associating
with each f a 3 a suitable functional defined on the convex
cone. Q. of functions 9s{ defined on E, real and 0, and 0
outside of a compact
48. I. Functional on Q+ associated with an element f a 3...
Let fa 3, .and let ? a Q+. For every number X > 0 let E be the
set of points x of E such that
The set Ez is a compact set which decreases when X increases,
with 'E c support of ?. Then/(E>.) is a positive, bounded
decreasing function of k. Set
where I(?)is the interval ]0, maxx].
When f(½),--.. 0, this integral. can be written as
f(?) ,, +f(E>,)dX -
In particular,
integral f? d,.
when f is a Radon measure, f(?)is simply the
48.
Immediate properties of the functional f(?).
Clearly f(?) :> 0.
f(,) _f(q) if ?, < % ;in other words, f(?) is
For every a > O, f(a?) a[()
increasing.
But conversely, each functional defined on Q. and posse.ssing
these three properties is not necessarily the ftinctional associated
with some element f a :i, We shall see later interesting exam-
ples of this fact.
48. 3. Re{lularization of ements f of 3. -- Denote by 3 the
convex cone of functions [ associated wth elements fa3.
The mapping f-- f of 3 into 3 linear. We shall investigate
the inverse image of an element f in this mapping.
For every [ a ,, the regularized [unction associated with [
s the function f,. on,,.(E) defined by
f.(K) inf f(x)
o
KX
(K and X a 3/.(E)).
266 usxvr cOVT
If W denotes any (( proximity ) of the uniform structure of
E (assOciated with any of its compactifications), and if Kw
denotes the neighborhood of order W of any compact subset
of E, then the above may be written
f.(K) lim f(Kw).
W--o
This form enables us to show that many properties o{, f are
preserved by regularization. For example, if f is sub-additive,
or alternating or monotone of order n or c, the same is true
of L.
It is immediate that f, is 0, increasing, continuous on
the right, .and is the smallest of the functions larger than f and
possessing these properties. In particular, for every,f 3 the
condition f f. is equivalent to the condition that f be
continuous on the right.
An essential property of regularization is the equality f .... f,
for e,ery f .
Indeed, for any ), > 0, we have f(Ex)f.(Ex); and
L(Ex,)-< f(Ex,)
for ),, < ),, since
Ex: c E L.
Then if. we set u- f(Ex) and u.(,) .---f.(Ex), it follows that
u.(k)- hm sup u(),') or u.O,)- the smallest decreasing fane-
tion greater than u0,) and continuous on the left.
Thus
The set of elements f of ' which are continuous on the
right is clearly a convex cone, which Will be denoted by ,.
The preceding shows that the canonical mapping of , into
is a mapping onto .
Let us sho,, that the canonical mapping f-,-f of 3 onto 3
one-to-one.
It is sufficient to show that for every [ a and for every
K 3{(E), f(K) may be determined when f is known.
Now let (K) be the characteristic function of K; then
f(K)- inf f(?).
'Indeed, f( K) <_. f(?) for every > +(K), and since f is conti-
nuous on the rght, for every > 0 there s a compact neigh-
THEORY OF CAPACITIES
267
borhood V of K such that (f(v)- f(K))< .. Now, since E
is locally compact, there exists a function ? a Q. such that
0 < ? .< , ? 0 Outside of V, and ? on K.
Then f(?)</(K) q- s, which proves the equality.
48. 4. Vague topology on 5. The set of elements f of
which have the same image f in is identical with the set of
elements f whose regularized function is the element f. of "
which corresponds canonically to f. In other words, if H,
is the canonical map of onto and H that of 3 onto 3, then
H HoH.
Let is be the topology of simple convergence on and
'gs the topology of simple convergence on . The in,erse
image iSv under of the simple topology s on 5 is called the
vague topology on 3.
In other words, a filter on 3 converges vaguely to an element
L of whenever, for every , a , the values f() converge
to f0() relative to this filter.
Or again, for every f0 a I, a base of vague neighborhoods
of œ0 consists of the V(s, (?)) where this symbol denotes the set
of i a ' such that
if(,) --fo(?i)!
(a_> 0; Q. with i I
and finite).
The map H of .v into is is continuous by construction. Let
us show that the map H of is into gs is also continuous.
This result follows immediately from the fact that, for every
filter on : which converges simply to an element [0 of 3, for
every , and every X, the f(Ex) converge to /0(Ex); then if
u(),) /(E), the u(k) converge to u0(k). Now u(Z) is decreasing
and thus
Thi s
fo(?).
statement s
uo(X).
equivalent to
saying that the f(q)
converge to
The restriction of H to ;. being a one-to-one map of 3.
onto 3, H defines a homeomorphism between 3.with the vague
topology and with the simple topology. But it must be
noticed that the restrictions of iSs and of v to 3. are not iden-
268
GUSTAVE CHOQUET
tical (except for very special cases). T ø prove this statement
it is sufficient to take a sequence f, of Radon measures on
E--[0, ] each consisting of a point mass q- at the point
x i/n; this sequence converges vaguely to f0 the point
mass.+ i at x--0, but it does not converge simply to any
function f0.
The spa. ce 3-with the topology iSv is not aisHausdorff space.;
th e associated Hausdorff quotient space homeomorphc
with or with 3, with the vague topology.
It can be proved, but we shall not.do so here, that w. it.h the
topology iSs, 3, is everywhere dense in 3; and that, smfiary,
if c (respectively &b) denotes the convex closed subcone of
consisting of the alternating (respectiv.ely monotone) functions
of order c, then (-,
48.5. Study of the case where E is compact. For every k > 0,
the set ,3(k) of all those f ,9 for which f(E).<___k is obviously
compact with the topology of simple convergence. Thus since
for every f0 3, the set of those f for which f(E)<__ 2f0(E) is
obviously a neighborhood of f0, , is locally compact with the
topology of simple convergence.
Now since the mapping H from is into gs is continuous, the
image H((k))is compact; but, since for every fa ':i _
,, f(E) f(l),
the set H(3(k)) is identical with the set of all those 5 for
which f(l) .< k; moreover, every f0(W-= 0) has as neighborhood
in the topology s the set of all those for which ([)< 20().
Hence, is locally compact.
It follows that 3, is locally compact in the vague topology.
The same holds for the sub-cones e% (and .., of 3, consisting
of all positive alternating (monotone) capacities of order
(Jb) on 3{.(E).
In 3, et, and .,%b, the subsets consisting of all functions f(f_--O)
which take no values other than 0 and I are obviously compact
(since f(E)' i); the same is true for the canonical image of
these sets into 3., c%, and .. Now, if f a 3, and if f takes
no values other than 0 or l, the same holds for the regularized f..
Thus, since these functions are the same, within a constant
factor,. as the normalized extremal elements of 3., c% and
THEORY OF CAPACITIES
269
those sets e(3.), e(et.), e(A'b,) are compact in the topology of
vague convergence.
48. 6. Study of the case where E is locally compact. The
topology s on 3 is not locally compact in this case, but it is
easily shown that 3 is complete under the uniform structure
associated with the topology of simple convergence. The
same is true for the closed sub-cones c and
Likewise, ,., ca.. and .rb are complete under the uniform
structure associated with the topology of vague convergence.
It may be useful to remark that, for every [0 3, the set
of all f_<_ f0 is compact under the topology of simple conver-
gence. (The same holds for a and .!b). This would still be
true if [,, were replaced by an arbitrary non-negative function
defined on 3.(E).
The same is true on 3 (also on e% and A.)-with the vague
topology.
The following is another restriction which leads to compact
sets.
Let
us set f((I))--sup f() ( a Q+) for every real-valued
non-negative function (I)which is continuous on E. Then the
set .of all [ a 3 for which [((I)) ___< k is compact in the topology
of smple convergence, for evrry constant k > 0. (The same
holds for ½x and Ab).
The above proposition holds also
vague topology.
on 3. (and c, dib.)
with the
48. 7. Extension of f(?) to non-ne(lative, upper semi-continuous
functions which vanish on th.e complement of a compact set. We
have associated f(9), defined on Q+, with every f
Let us designate by SS. the set of all positive upper semi-
continuous real-valued functions ?(x) defined
vanish outside of some compact set.
Furthermore, for each ,o a SS+, set
on E which
f(?0) - inf
(? Q+).
The notation f(?0) is consistent since, if 0 a Q+, the extended
function [ takes the same value as the function which was
270
originally defined on Q+. Thus, ;ve have indeed obtained
an extension of f.
We shall henceforth assume that [ is continuous on the
right, that is, f a 3; in other words we shall assume that f is
a positive capacity on .(E). Then we have [(,) ..... [(?) whene-
ver ? s the characteristic function of a compact set.
More generally, it is easily verified that, for every ? SS+,
fq?) '.. /[(:)/(E>,)dX,
where Ex agmn denotes the compact set of all points x for
which
The lower integral of every positive function defined on E
(relative to œ) can then be defined by the classical procedure.
In particular, this lower integral is defined for every positive
log, er semi-continuous function defined on E. The upper inte-
gral of every positive function ? on E can then be defined as
the infimum of the upper integrals of all lower semi-conti-
nuous functions greater than ? on E.
Hence we have a concept of a capacitable function. It
would not be very difficult to extend this concept to functions
of arbitrary sign on E.
In order to obtain significant theorems, it would be necessary
to place certain restrictions on the function [, such as, for
instance, that f be alternating of order 2.
49. Intetral representation of the non-neõative capacities of
order cx on .(E). We make the initial assumption that E
is a Compact space. The cone et of all positive capacities of
order eL. on 3'(E) is therefore locally compact in the vague
topology, and the set of its normalized extremal elements is
compact. Let us recall that these normalized extremal
elements are the functions fT(X)defined by
f,.(X)!0 for XT ',
t for X T =/=
where T is an arbitrary
compact subset of E.
Let ; be the set of these elements/T(X)
The vague topology on g,(distinct from the topology
even on this subset of (,), which may be considered as a
THEORY OF CAPACITIES 271
'topology on the set of all elenents T of .(E), is identical
with the classical topology of .(E).
For we have for every T a 3.(E), and each ? a Q+,
fT(9) '--" max () on T.
It follows immediately that for every
converges to T,, in the classical to
every 9 to fTo(?).
ConverSely, assune that
filter on ..(E) which
pology,/a,(?) converges for
for some filter on ;;'.(E)the b(9)
converge to /T(?), and that (o,) is a finite covering of To by
open sets each of which meets To.
There exists 70 a Q+ with 0< 70<, such that 70--" on
[ (U(0,)) and 0 0 on To. For every i, there exists ?, a Q,
with 0 :.< 1, such that ?---- 0 on I (o) and nax (?) on
Tois .
Hence there exists a set belonging to the given filter such
element T of this set is contained in [.J (o) and meets
Thus this filter converges to To in the classical
that every
each
sense.
Hence, in view of the general theorem (see 39.4.) there
exists for every [ae% a non-negative Radon measure I on
:i(E) such that
f(?) (9) d. (T) for every
This formula may be extended to every 90 a SS+. For, such a
9o is the limit of a decreasing filtering set of functions ? a Q+.
Hence, f?0)is the limit of the (?) with respect to this filte-
ring set. On the other hand, fr(?0) --: (max (%) on T) is the limit
of (max(?) on T) with respect to this filtering set. This
function fT(9o) is upper semi-continuous on ..(E), and its inte-
gral f(?o)dy.(T)is indeed the limit of
In particular, if for ? we choose the characteristic function
of a compact set X E, then
f(x) , fF(x) d(T)
for every compact set
XcE.
In other words, f(X) is the -measure of the set of all compact
272 GUSTAVE CttOQUET
sets T which meet X. Thus, the capacity/(X) may be obtained
from the fundamental scheme (E, F, A, ,), where E is the
given space, F .(E) with the classical topology, A is the
set of all points (x, X) of (E X' F) for which a X, and y. is
the Radon measure on ,.(E) which we have introduced above.
We had previously established that the set functions obtained
from a Radon measure by neans of a finite number of U-ho-
momorphisms are capacities of order et. We have now pro-
ved the converse.
More precisely, we can state the following theorem.
49. t. THEOREM. Suppose that A, is the smallest class of
all real-½,alued functions [ each of which is defined on the set
of all compact subsets of some compact space E, such that the fol-
lo,ing conditions are satisfied:
(1) & contains e½,ery non-negati½,e Radon measure defined on,
any compact space E.
(2) If E and F are tg,,o
U-homomorphism ,hich is
compact spaces, if Y = (X) is a
continuous on the right from
into :K(F), and if f & is definedon 3.(F), then e ha,e e(X) a .t,
½,,here e(X) is the function defined on 3.(X) by e(X)/(9(X)).
This class db is identical ,,ith the class of all positive capacities
of order eq, defined on the sets .q{.(E) relati½,e to any compact
space E.
49. 2. Probabilistic interpretation of this result. ., We have
already, in particular cases, interpreted the scheme (E, F, A,
as a probabilistic scheme.
More generally, let such a scheme be given, in which E and
F are two abstract sets, , a simply additive positive measure
defined on an algebra of subsets of F ;vith Iz(F) l; and
let us denote by g an additive class of subsets of E such that
for each X a the set Y (X) obtained from X by means of
the construction of 26. 8, Chapter v, belongs to
We know that the function /(X)- (Y) is alternating of
order oc on t.
Now let us consider , as an elementary probability on the
set F of events. Let us consider E as another set of events,
and A as the set of all faporable encounters (x, y) with x a E and
THEORY OF CAPACITIES 273
ya F. Then f(X) is obviously the probability that such a
favorable encounter occurs at least once on the subset X E.
Conversely, the preceding theorem shows that, if sufficient
cond. itio. ns of regularity are imposed on E, g, f (compactness,
contlnmty on the right), then every positive function of X
which is alternating of order infinity expresses the probability
that a favorable event occurs at least once on X.
Actually, the regularity need not be of such strong form.
And the fact that the set of all non-negative functions which
are alternating of order infinity on an additive class of sets has,
as extremal elements, functions whose values are 0 and 1,
shows that one could certainly always interpret such a function
in
as.a probability; but. it would undoubtedly be
ths case to generahze the notion of additive
on F.
necessary
measure.
Whenever one can prove that any function defined on the set
:(E) of compact subsets of a compact space is a positive
capacity of order eta, one is sure that it could be interpreted
in terms of probabilities.
In the most interesting cases (such as the theory of poten-
tial), the space E is not in general compact, but only locally
compact, and the function [ is not bounded from above; hence
it is not possible to give a direct probabilistic interpretation
of f.
However, the brief study of the ease where E is locally
compact which follows in section 49. 5. will show that the fun-
damental scheme still exists in this case, and that it is therefore
possible to give (( locally an interpretation of f in terms of
probability theory. If in particular f.s bounded øn ..%'(E),
then it is sufficient to divide f by sup f m order to obtain the
desired probabilistic interpretation.
49. 3. ExaMrnr. If E is a Greenian domain in the space
R" and P a fixed point of D, we denote by/(X) the harmonic
measure, for the domain (D X), of the compact subset X
of D with respect to the point P. (f(X) I if P X).
We know (26. t2, Chapter v) that œ(X)is a capacity of
order ct on :;(E), and that 0.<_f<_ 1. Hence f must admit
an interpretation in terms of probability. That interpretation
is known (see Kac [1 and 2]); f(X)is the probability that a
274 G USTAVE CHOQUET
particle issuing from P and undergoing a Brownian motion
will meet X at least once before it meets the boundary of D.
The support of the measure ?. is in this case the set of all
supports of Brownian trajectories issuing from P and contained
ß D
In this particular case/(X) can be extended to the set of all
compact subsets of the boundary of D; its restriction to the
set of these conpact sets is then a Radon measure, which is
identical with the ordinary harmonic measure. Obviously, the
boundary can be topologized by various topologies which lead
to diverse harmonic measures used
ramified, geodesic, and Greenian
Choquet []).
in modern potential theory:
measures (see Brelot and
49.4. Expn. -The Newtonian or Greenian capacity
F(X) of a compact subset X of a doandin E admits a less
simple interpretation; this situation is due to the fact that
/(X) is not bounded on ;.(E)(see Kac [2]).
49. 5.
compact.
but not
from E
Inteõral representation in the case where E is locall
Suppose that the space E is locally
compact, and that E s the compact space
by adjoining the point o. The locally
compact,
obtained
compact
topological space [.() lo!], where. loci is. the element
of ;.(E) consisting of the smple point ½o, s isomorphic
with the set 5(E) of all non-empty subsets of E with a
suitable topology. When we shall talk of (E), it will be
understood that that topology has been placed on .7(E).
We have already shown that the extremal elements-of the
cone c, of the capacities of order a on .i(E) are the func-
tions/T(X), where T is a non-empty closed subset of IV., with
fr(X) !0 if (TflX)--p'
--!l if (Tf X)--/::.
For every [ cx,. and each compact set K E, let us denote
by [. the capacity defined on :;.(E) by h(X)--/(Xlq K).
There exists a measure , defined on the compact subset
:;(K) of 5(E) and corresponding to h, such that
h(x
. .(X fl K)dp. (T) -- / [T(X) d?. (T).
THEORY OF CAPACITIES
275
Using the facts that f(X) < f(x) and that f(X)= lira f(X)
with respect to the set, filtering on the right, of all compacts
K, one can show that the measures converge vaguely,
with respect to this same filtering set, to a measure on ,(E),
and that
f(x) ff(x) (T),
for every
x.
We have again in this case, for every a SS+,
f(?) d, (T).
The relation f d?. c holds if and only if f is bounded; in this
case the preceding formula is valid for every upper semi-
continuous non-negative q on E.
50. Inteõral representation of the non-neõative capacities of
order .,llb on 5.(E). Uniqueness of this representation. The
reasoning and the results in this case are closely analogous
to those pertaining to the capacities of order et©; when E is
locally compact, the proof and results are even simpler than in
the case of the capacities of order et=.
Let us first suppose that E is compact. The extremal ele-
ments of M are the functions /T(X) (where T is a compact
subset of E) defined by
o if Tel-X,
I if TcX.
For every f .11¾, there exists a measure I . 0 on Y;(E)
such that
[(?) :-- ffT(?)dl (T) for every ß a SS..
In particular, if we take for ? the characteristic function of
a compact subset X E, we see that œ(X)is the ?measure
of the set of all T for which Tc X.
Hence the following geometrical interpretation: :.(E) is a
compact space, ordered by inclusion. For each XaS(E),
the set of all T X is a compact subset of .(E); we shall call
this set the negative cone with vertex X in .;.(E).
For every f A./o there exists a measure 0 on ,; (E) such
that we have, for every X a ;,(E).
f(X)- the -measure of the negative cone of vertex X.
276 GUSTAVE CHOQUET
We have thus in a particular case a new proof of a general
theorem obtained by A. Revuz [3], which furnishes a simple
integral representation of all ½½ totally monotone functions
defined on a partially ordered set S (here S--; (E)), when cer-
tain conditions of regularity are satisfied.
The functions studied by A. Revuz are identical with the
functions monotone of order (q> 0), defined on a semi-
group consisting of an ordered set S on ;vhich the semi-group
operation is the operation (a b) which is assumed to be
always possible.
The general theoren of section 47. 2. shows that the cone of
these functions admits as its only extremal elements exponen-
tials (which take no values other than 0 and 1, since the semi-
group is idempotent), because there are no extremal elements
in S (we have x x -. x for every x).
Now the set of all points x in S where a given exponential hb (x)
takes the value t is invariant under the operation -, and it is
hereditary on the left;conversely, one may associate with each
subset of S having these properties an exponential whose value
is I on that subset and 0 elsewhere.
It can thus be foreseen that, if every negative cone of S is
compact, then it is possible to associate with each exponential
a point P(') of S such that (x) equals I or 0 according as
x>.P(') or not.
I follows that in cases of sufficient regularity there exists
a representation of totally monotone functions f on S by means
of measures > 0 defined on S and such that:
f(x) .... : the -measure of the negative cone with vertex x.
The very subtle analysis undertaken by A. Revuz enables
him to show the uniqueness of that measure in general cases.
In particular, this measure is unique when S is the ordered
set :;'(E) associated with the compact space E, in other words,
if we are dealing with capacities of order Al% on S.(E).
This uniqueness makes it possible to extend these results imme-
diately to the case where E is an arbitrary Hausdorff space.
More precisely, we have the following theorem.
50. t.
and f a
THEOREM.
non-negati,e
If E is an arbitrary Hausdor]7 space,
capacity of order A% on (E), then there
THEORY OF CAPACITIES
277
exists one, and only one, generalized, non-negative Radon
measure , (see 26.6, Chapter v) defined on .,(E) ith the classical
topology such that, [or e,erg compact set X E,/(X) is the -mea-
s?zre of the compact negative cone of ,ertex X in ,q;.(E).
To prove this extension, it is sufficient to observe that, for
every compact set Xc E, the restriction of œ to .(X) is asso-
ciated with a Radon measure whose support is ..(X), and that,
if X, X, then the measures thus associated with X, and X_ are
compatible on ..(X,) because of the uniqueness of these
measures.
50. 2. Probabilistic interpretation of the elements f of
We have already remarked that the probability fthat a favo-
ruble event occurs at least once on a set X c E is a function
of X which is alternating of order c; thus, the function
g(X) -- [( C (X)), which expresses the probability that
this favorable event occurs never on the complement of X,
is a function which is monotone of order c.
Conversely, the above result shows that, under the condi-
tions of regularity which we have indicated, and if, moreover,
E is compact and œ(E) , then each function/(X) which is
monotone of order expresses the probability that some favo-
rable event never occurs on the complement of X.
51. Uniqueness of the representation of a non-negative capacity
of order t= on ,.(E). Suppose that E is compact, and that
f is a non-negative capacity of order et on ,%.(E), and let be
one of the Radon measures on .%.(E) associated with f.
Let , of order ,!b, be the conjugate capacity of f (see t5. 6.
Chapter
If we set g .=/(E) q-/ then the capacity g is non-negative
and of order Abe; hence, a uniquely determined Radon
measure > 0 on .(E) is associated with g.
Now fo' every compact set Xc E we have
/(X)--the -measure of the set of all T such that X fq T--/=0;
hence: g(X) -- f(E) + -- f(E) f(E X)
is the -measure of the set of all T which do not meet (E X)
and which are therefore contained in X.
278
Since . is unique,
theorem.
GUSTAVE CHOQUET
we have
hence the following
51. . Tnsoas.. For e,ery non-negati,e capacity f of
order a on..(E), here E is compact, the measure on ..(E)
associated ith œ is unique ; [urthermore, i[ denotes the measure
on .{E) associated ,ith the non-negati,e capacity go[ order
defined by g- (E)q- f, then e ha,e . v.
We remark, without giving the proof, that this result can
be extended to the case where E is locally compact, in the
following form:
(1) ' '
s umque;
(2) heneer
g - (f(E) + f) is defined, that is,
whene,er œ is bounded.
When [ is not bounded it is still possible to define a function
associated with [ by using the following definition:
g(X) -- the -measure of the set of all T X.
It can be shown that g(X) is the limit, as K tends to D, of the
functions
g (x) := f(K) + ,, (x) -, f(x)
x).
For example, suppose that œ is the Greenian capacity relative
to-a domain D; it can be easily shown that, for every X, we
have
[f(K) -f(K X)]-+ 0 as K- D.
It follows that g(X)- 0; this fact implies that the measure
on $(E) has as its support the set of those closed subsets of D
which are not compact.
52. Functional study of the elements of et and .b.
defined, for every f ,3 and for every a Q,
We have
As we know, it follows that f is positive, increasing, and
positively homogeneous. ^
We now seek to establish what can be said about [ when
certain restrictive hypotheses are placed on [, such as, for
THEORY OF CAPACITIES 279
instance, that f be sub-additive, or that f belong to a., or that
[ belong to ,.
Let (E X R) be the product of E and the real axis; and for
every 9 Q+ define [?] as the set of all points (x, y)in
(E X R) such that y <___ 9 (x). Furthermore, let (),) be the set of
all points (x, y)
obviously,
(E x R) such that y >
E;, . n'.(x)).
We have,
Now 'the following two relations are true:
---- u
[, ,--, %] ,, [?,] n [?,],
where- and denote the operations sup and inf on the ?.
These formulas nake it possible to transform every relation
satisfied by œ, which involves no operations other than inter-
section and union, into a relation satisfied by f(Ek) and involving
the operations and o.n the
If these relations are hnear, tlen it is possible to integrate
and to obtain relations satisfied by œ. If, in particular, œ is
sub-additive, then
q-
If f a ex, then f is alternating of order o on Q+ (relative to
the operation , on Q+).
If/a.b,.then f is ,nonotone of order c on Q+ (relative to
the operation - on Q+.
But it is not true that every functional on Q+ which is non-
negative and increasing, and which satisfies one of the three
preceding conditions is identical with the o/f associated with
an œ a , where œ is respectively sub-additive, order c or 1,.
This will be shown by examples, in which we shall choose
E such that E- 2.
52. J. Study of an example. ,,- Let the points of E be x,
and x. Every function ?aQ. is defined by its values
// -- ?(x) (i , 2). Thus Q+ is isomorphic to the ordered
cone R'-' of all couples (/, /) Every [a 3 such that [(½) 0
19
280 avsvœ COQV.
is characterized by the three values f(x,), f(x), and f(x,, x).
To say that [a . is the same as saying
f(x,) o, f(x) > o,
f(x,, x,) > sup (f(x,), f(x)).
To say that f a c is the same as saying
f(,) > o, f(,) k o,
sup (f(x,, f(x,)') _< f(x,, x,)< f(x,
1+ f(x,.,).
To say that f a ,.,t'b is the same as saying
f(x,) > O,
f(x,) + f(x,)
f(x,) O,
f(x,, x,).
In each of these three cases, the function f(9) - f(y,, y,) is a
linear function in each of the regions y,__.y, y<_y,; it is
defined by its values on the lines y,- 0, y--0, y, y and
so it depends upon three paraneters.
Each function which is. not of this type cannot belong to 3.
The following are three such functions on R which are more-
over increasing and positively
alternating of order for the
order c, for the operation -:
homogeneous
operation
and
and
respectively
monotone of
x - + + y' xy
, , or
g '" x+y g x+y
g
53. Definition and properties of the classes I, A, M. Let E
belocallycompact. Wedenoteby I the cone of the functions /(?)
defined on the lattice cone Q+ which are (a) positive,
(b) increasing, and (c) positively homogeneous.
We denote by A (respectively M) the subcone of I nade up
of the functions on Q. ;vhich are alternating of order c for
the operation (monotone of order c for the operation -).
We k.ow already that ,3c I; et._...A; ,]bcM. When E is
compact, these cones I, A, M, are locally compact under the
topology of simple convergence.
We can easily extend the definition of each [ belonging to
one of these classes to the lattice cone SS+, with preservation
of th e functional properties of [.
We shall no;v state without proof several results about the
structure of these cones.
THEORY OF CAPACITIES
281
53. I. Extremal elements of A. ,, The extremal elements of
A are the fu. nctons f O, positively homogeneous on Q+, and
such that
(f(?,) .. f(?,))== (f(?, ,.., %) -- .-
An equi,alent condition to this is the [ollos, ing :
sup (f(?,), f(?,)).
It is immediate that each such function belongs to A, and
that it is an extremal element of this cone. The converse is a
little more difficult.
These extremal elements can still be characterized in an-
other way. Let (x)½ be an arbitrary family of points of E, and
let (),)½ be some constants > 0. The functionf(?) sup k?(x),
which is assumed < q- for each ?, is an extremal element;
and conversely each extremal element is of this form.
This last formula can also be written as
f,(9)- max (?(x).(I)(x)),
xE
vhere ½P (x) is any function .> 0 and upper semi-continuous on
E. There is a one-to-one correspondence between the extremal
elements of A and the functions to which they are associated.
For example, if (P t, f(x) is the ordinary norm on
When E is compact, it is immediate that each [ a A admits
an integral representation such that
f(?) - for each ? a SS
where I is a measure on the compact set of all normalized by
the condition
f,(l). I
or max ((D (x)) -- I.
The topology on the set of these ½I is by definition the
topology of sinple convergence on the corresponding [,.
This topology can be interpreted as follows: each ½I is repre-
sented in E x R by the compact set [(I] of points (x, y)
where 0_< y ((x). The set of these [(I)] is a compact subset
of the space 3(E X R) of subcompacts of E X R. The topo-
282
GUSTAVE CHOQUET
1ogy thus induced on the set of normalized (1 is identical
with the preceding topology.
The measure associated to each f a A is unique.
Example of elements of A:
f()
Iføreach a > - and
each measure v0 on E.
53. 2. Extremal elements of M. We obtain a characteri-
zation of the extremal elements of M analogous with the
preceding by changing the operation to , and sup to in[.
We can write them in the form
f (?) -- inf k,? (x,).
(We will have f 0 only if the-(x,),½, are taken on a compact
Set.)
Or else, by designating by an element of SS+ (hence zero
outside of a compact),
f+(?) .-- ,nax. of numbers k :k: 0 such that k+(x) ?(x) for each
x a E, which amounts to saying that
with the convention
. (x)
f+(t)--m,
() ..... , +
There is a one-to-one correspondence
and the extreme elements of M.
o.
between the , a SS+
For example, if E is compact and if '1, we obtain
f+ minimum of ? on E.
Example of an element of M.
f(?)--
for each = t/p with p a positive integer, and
v0 on E.
The normalized set (by ,nax + ..--1)of 4/a SS+ is
compact by the topology of simple convergence on the
for each
locally
corres-
THEORY OF CAPACITIES
283
ponding [. For each f a M there exists on this set one and
only one measure , such that
f(t) -
for each ? a SS..
In other words
f(?) =/"ds.',
which is 0 only when has its support contained in the
support of 9-
Let us remark that, since f(?) is a function 0, monotone
of order o and continuous on the right on the lattice SS.,
there exists, according to the theorem of A. Revuz mentioned
previously, one measure v > 0 and only one on the locally
compact space SS such tha t
f(?) ,,.
Y-measure of the set of ?'< 9.
The measure , is obtained from ¾ by the following relation:
(A) = ,,,(B)
where A is an arbitrary compact subset of the set of normalized
: and B is the set of ' SS+ of the form
,' 0, where 0 ..< 0 ! and ' a A.
Conversely, , is also determined as soon as , is known.
53. 3. Extremal elements of A rl M. The elements f of
A n M are characterized by the following relations:
(-)
(c)
f>o;
f(X?)--xf(l o x.v.o;
f/, )+ f(? %) = f(,) + f(/.
The extremal elements of the cone A fq M are, up to a constant
factor, the/(?) ?(a), where a a E.
For each œ a A fq M there exists a unique measure ,.. 0 on E
such that
f(t) j"f(?) d (a) - j",(a) dp..
In other words, the cone A rlM is identical with the cone
of Radon measures on E.
284 GUSTAVE CHOQUET
53. 4. Study of the cone I. We want to show that the
elements of I are closely related to the elements of A and M and
more precisely with extreme elements of these cones.
53. 5. Tssoas. (a) The superior cue, elope (supposed
finite) and the in[erior end, elope o[ any [amily o[ [unctions [()
belonging to I also belongs to I.
(b) Any element of Ils the superior (in[erior).end, elope of
a [amily o[ extremal elements of M (respectin, ely A)
Proof. (a) The first part of the theorem is immediate since
homogeneity and monotony are preserved by the operations
sup and inf.
(b) Now let f a I; for each 90 a Q. where 70_=/_0, the function
f,o(?) - min/o(X),
with the convention of section 53. 2. is an extremal element of
M; the same is true of
g =
Now, for each ?.
In fact, if we set
.. [?o(?)- min (,-oo),
we have ? :> X?o and' hence f(?)> X[(?o),which is exactly the
required relation.
Hence, not only is [(?) the superior envelope of a family
of extremal elements of M, but for each ?0 there is one of these
elements, namely/(0)f:,(), which is equal to f(9) for ?- 9,,.
(c) For each > 0 and for each 70 a Q (with 7o_0),
let (x) be a continuous positive function on such that
(x) <s and let ? ... sup (?o,(x)).
Let us show that
where
f(?,) - sup f(?') all
THEORY OF CAPACITIES
285
In fact, if
), ---. max (9-...),
we have ? <),? and so
required equality.
Now if is small enough,
hence,
max
which is exactly the
which is an extremal element of A, takes the value f(9) for
?-- %ø
Hence, if for each % and each s we can choose s(x) such
that ([(?). /(90))is arbitrarily small a restriction that we
have not indicated in the wording of the theorem in order not to
complicate it we have proved the last part of the theorem.
This restriction, in the case where E is compact, concerns
only functions 9 which can take zero values on f.E; it is equi-
valent in a way to the continuity on the right of Note here
that when E is compact, or more generally when E is the denu-
merable union of compacts, this restrictive condition is satis-
fied for each œ a I which is sub-additive (for the operation ),
for example for each [ a A.
53. 6. Extremal elements of I. When 1 is finite, we can
give a complete characterization of the extremal elements of I.
The study of .an [, f f(y,, yo.,:.. y,), of I amounts indeed to
the study of ts trace on the smplex z defined by y 0 and
Y,y .-i. This trace is locally Lipschitzian on the interior of
the simplex; in order that this be the trace of an extremal f, it
is necessary and sufficient that almost everywhere the graph of
this trace has a tangent hyperplane which passes through
any one of the n faces of the simplex.
For example, and this is valid for any space E, each [ which
is the superior or inferior envelope of a finite family of ele-
ments of A or M is an extremal element of I. From this it
286 avsAw COQV
follows that the set of extremal elements of I is e,eryg,,here
dense on I.
53. 7. Primitive elements of I and the operations. sup, inf,
and _f. ,,. Let us call each multiple f (where O)of
the fu nction f.(½) (a) a primitive elementof I.
The preceding shows that we can generate A, M, A flM,
and I by starting from the primitive elements and applying the
following operations: superior envelope and inferior enve-
lope of a family of functions f, and th e operationfft d,(t),
where [ is a non-negative measure on a set of elements of I.
More precisely, let 2 be the class of primitive elements
(which are indeed the extremal elements of A fq M).
The class a". of elements obtained from ' by the operation
sup is made up of the extremal elements of A.
The class ' .
.%.. is made up of the extremal elements of M
The class 2fis identical with A M.
and '" identical to I.
The classes .x,v, .f ,f, . are
The classes
and M.
and œt,, fare identical respectively to A
'?' / ) is identical to
One could show that the class f.,.r k J,
the class of positive, increasing, positively homogeneous and
' S
V-sub-additive (V-super-additive)(') function defined on Q+.
It would be interesting to characterize also these classes in
terms of the operations and .
For example, we can see easily that if fzf .o' we always
have
2f(?, ,, ?) f(?, ,- ,,) + f(?. ,) + f(?, ?,)
as well as other inequalities of the same type.
Let us add that we cannot form, with our three operations,
classes other than those which we have pointed out above.
() For the definition, see section 56.
THEORY OF CAPACITIES
287
53.8. REMARK. "We obtain an analogous classification
of the positive and increasing set functions by using the ope-
rations sup, inf, and f.
For example, if we consider the functions œ(X) defined on
the set 2 s of all subsets of .a set E, the primitive functions are
the functions œ(X), where u is an ultra-filter on E, with
f,(X)--10
if X a ultra-filter
otherwise.
O'ur three operations lead to the classes a f'lJb, ex, A.'b, ,,
and to other classes which we have not studied.
Various problems related to these operations could be consi-
dered. For example, if we apply the operation sup or inf to
a family of positive capacities on the set i(E)(where E is
compact), for which the K-borelian sets are capacitable, to
what extent is it the same for the function thus obtained?
54. Relation between the alternatin{l functions of order 2 and
the pseudo-norms. Let us first,prove a lemma relative to
the V-sub-additive. or V-super-additive functions on a
Let ½ be a lattice cone, that is, a convex cone. such
its..natural order structure is. a lattice structure.
A real function [ on c? is called V-sub-additive
additive) if
cone.
that
(V-super-
() f(x.) Xf(x) for each X > 0;
(b) f(a b) f(a) f(b),
(respectively
54. . THEOREM. If the [unction f
homogeneous and if it satisfies
OFt ( 1;8
positively
or
(2)
fia , b) + f(a b)<5 f(a) + f(b)
f(a b) + f(a b)__f(a) 4-,. f(b),
then f is respecti,ely V-sub-additi,e or V-super-additi,e.
Proof. The proof is based on the proof of the special case
where t is finite dimensional (hence isomorphic to RS), and
where œ possesses continuous second derivatives for x =/= 0.
Let
f(x) f(x,,..., x).
288
GUSTAVE CHOQUET
If
...,
b --Ix,, (x: + h:), ..., (x, + h,)l , where h i O,
we have
When hi O,
condition (!) implies that
hence,
and, more
for i --f- l.,
generally,
Now since f is homogeneous of order l, we have
Y xf, :: 0, for each i.
.Therefore the terms F of second degree in the development of [
n the neighborhood of the point x satisfy
,, dx . O.
2F: x,xL, (x) ! ' x,
It ollows that f is locally convex on ½ and hence also globally,
and this is known to be equivalent to saying that f is V-sub-
additive. For the V-super-additivity, it is sucient to change
f into f.
Let us notice that the converse of that theorem is false.
For example, in ½. R the function
+ y)
x-4-y+z
is V-super-additive (and it is increasing also), but it does not
satisfy the inequality (2). The function
z- +x+y
f' x +y+z
!s V-su.b-additive and increasing, but it does not satisfy the
lnequahty (t).
In order to verify this, it is sufficient to take a .-- (0, I, t)
and b .-- (t, 0, 1).
THEORY OF CAPACITIES
289
54. 2. APPLICATION. - Let E be a locally compact space, f
an element of :,, a.nd .the function on Q+ which is associated
to it. Recall that [s sad to be a pseudo-norm on Q+ if we have
() f(?, + ?,)=f(?,)+
TtgORgM. In order that be a pseudo-norm on Q+, it is
necessary and sucient that [ be 'an alternating capacity o[
order .
Proof.-S. ince f is increasing, it is equivalent to say that f
is alternating of order ex, or to say that we have
f(X, U X,.) -I- f(X, rl x,) f(x,) --1- f(x,).
Now this relation is equivalent to
+
f(9,) -!- f(?,).
According to the preceding theorem this relation implies that [
is a pseudo-norm. Conversely, let us assume that f is a pseudo-
norm. It is immediate that the relation (i) above can be
extended to the functions ? a SS+. Therefore, if , and ? are
the characteristic functions of the compacts X, and X., we
have
fq9, -!- 9,) f(?,) -!- fqP,) -- f(X,) -I- f(X,).
Now (?, q- ?..,) -- 2 on (X,.f'l X,), is >..l .on X, U X¾ and 0
elsewhere. Hence by using the definmon of section 48. I.:
(?, 4- %)-= f(X, U X,) + fiX, 0 X,).
The desired relation follows immediately.
In the same way, we could prove that in order [or a positive
capacity [ to be monotone o[ order .:b..,, it is necessary and suffi-
cient that the associated [unCtion f satis[y the relation
An immediate application of this. theorem is the following:
If a capacity f is only sub-additive, its extension f is not
necessarily a pseudo-norm.
290 csoqrr. T...
BIBLIOGRAPHY
This bibliography contains several works which have not been referred to
in the text, but which are closely related to the theory of capacities and will
be useful to anyone working in this field.
AxssEN, Michael.
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362 (t953).
Department o!
BXlKHOF, Garrett.
[t]. Lattice Theory.
cations, vol. 25, revised
New York, N.Y., 1948.
BocHrEa, S.
ARONSZAJrq, N., and SMITIt K. T.
[t] Functional Spaces and Functional Completion. Technical Report t0.
Mathematics, Lawrence, Kansas, t954.
American mathematical Society Colloquium
edition. American Mathematical
Publi-
Society,
Completely monotone functions in partially ordered spaces. Duke Math.
J. 9, 59-526 (i942).
BOURBAKI, N.
[t] Elements de mathmatique. II. Premi/re partie: Les structures fonda-
mentales de l'analyse. Livre III: Topoloe gnrale. Chap. x: Struc-
Hermann et C ie,
tures topologiques. Actualitds Sci. Ind., nO 858.
Paris, t940.
[2] Elements de mathmatique. X. Premiere partie: Les
structures fonda-
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Hermann et C ie, Paris, 1949.
[3] Elements de mathmatique. XIII. Premiere partie: Les structures
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[4] Elements de mathmatique. XV. Premiere partie: Les structures fon-
damentales de l'analyse. Livre V: Espaces vectoriels topologiques.
Chapitre x: Espaces vectoriels topologiques sur un corps value. Cha-
pitre n: Ensembles convexes et espaces localement convexes. Actua-
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BRELOT, Marcel.
[t] La thiotie moderne du potentiel. Ann. Inst. Fourier, Grenoble 4 (952),
ti3440 (i954).
BRELOT, M., and CaQr., G.
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TABLE OF CONTENTS
INTRODUCTION
Chapter
Chapter
Chapter xiI.
Chapter
Chapter v.
Chapter ¾I.
Chapter
Borelian and analytic sets in topological spaces..
,- Newtonian and Greenian capacities .............
ß Alternating and monotone functions. Capacities ..
Extension and restriction of a capacity ........
Operations on capacities and examples of capa-
cities .....................................
Capacitability. Fundamental theorems ........
Extremal elements of convex cones and integral
representations. Applications ...............
Pages.
t38
t46
169
179
192
2i8
237
CHArTES I. BORELIAN AND ANALYTIC SETS IN TOPOLOGI-
CAL SPACES
1. Introduction ...........................................
2 Classification of K-borelian sets
3 K-analytic sets
4 The K-borelian sets
5. The operation of projection ...............................
138
138
t38
139
142
144
CHARTER II.
9.
NEWTONIAN AND GREENIAN CAPACITIES...
6. NewtonJan and Greenian capacities .......................
7. Successive differences ....................................
The inequality (/2)s < 0 .......... .......................
Complete system of inequalities .
Inequalities concerning all operations of the algebra of sets ....
Possibilities of extension of preceding theorems .............
t46
146
t49
t55
t56
165
166
CHAPTER III. ALTERNATING AND MONOTONE FUNCTIONSß
CAPACITIES ........................................
i2. Successive differences of a function ........................
t3. Alternating functions ....................... . .......... ..
i4. Set functions ..................................... . .....
t5. Capacities...... .................................... ....
t69
t69
t70
174
294
CHAPTER
t6.
17.
t8.
t9.
20.
GUSTAVE CROQUET
IV. EXTENSION AND RESTRICTION OF A CAPA-
CITY ................................................
Extension of a capacity ..................................
Invariance of the classes , by extension ...................
Invariance of the classes 111o by extension ...................
Extension of a class g, by a limit process ...................
Restriction of a capacity .................................
Pages.
t79
179
185
188
t89
CHAPTER V. OPERATIONS ON CAPACITIES AND EXAMPLES
OF CAPACITIES .....................................
2i.
22.
23.
24.
Operations on the range of capacities ......................
Change of variable in a capacity ...........................
Study of u-homomorphisms continuous on the right .........
Study of a-homomorphisms continuous on the right .........
25. Construction of alternating capacities of order 2 .............
26. Examples of alternating capacities of order ct .............
27. Examples of capacities which are monotone of order 11b© .....
192
i92
t94
t95
t97
t99
204
CHAPTER
VI. ,, J CAPACITABILITY. FUNDAMENTAL THEO-
REMS ...............................................
28. Operations on capacitable sets for capacities of order c% .....
29. A capacitable class of sets ................................
30. Capacitability of K-borelian and K-analytic sets .............
3i. Capacitability for the capacities which are only sub-additive.ß
32 Capacitability of sets which are not K-borelian
33. Capacitability of sets CA .................................
34. Construction of non-capacitable sets for each sub-additive
capacity .............................................
35. Intersection of capacitable sets ............................
36. Decreasing sequences of capacitable sets ...................
37'. Application of the theory of capacitability to the study of
measure .............................................
38. The study of monotone capacities of order vl/b,a ...............
218
2t8
221
223
224
225
226
228
229
229
230
23i
CHAPTER
VII.
AND
TIONS
EXTREMAL ELEMENTS OF CONVEX CONES
INTEGRAL REPRESENTATIONS. APPLICA-
39 Introduction
40. Extremal elements of the cone of positive increasing functionsß
4t. Extremal elements of the cone of positive and increasing valua-
42. Application to the integral representation of simply additive
measures... ............................. . ..... ........
43. Extremal elements of the cone of positive functions alterna-
ting of order o on an ordered semi-group .... ...... ...... .
235
235
240
242
245
249
THœOHY OF CAPACITIES
295
44. Topology of simple convergence on (z. Application . .........
45. Integral representation of the elements of cq ................
46. Extremal elements of the cone of monotone functions of order
on an ordered semi-group... .........................
47. Alternating or monotone functions of order on an arbitrary
commutative semi-group ......................... .. ....
48. Vague topology on the cone of increasing functions ...........
49. Integral representation of the non-negative capacities of order
cx. on .........................................
50. Integral representation of the non-negative capacities of order
Aooo on (E). Uniqueness of this representation ........
5i. Uniqueness of the representation of a non-negative capacity of
order coo on ,,(E) .............. . ....................
52. Functional study of the elements of c and b ...............
53 D finiti d p ti f th cl I A M ..............
. e on an roper es o e asses , ,
54. Relation between the alternating functions of order 2 and the
pseudo-norms .........................................
BIBLIOGRAPHY ......................................... . .......
Pages.
255
258
260
26!
265
270
275
277
279
28O
287
290
2O