ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 1991 MS Classification: 28 Zakharov, Valeriy, S. -Petersburg University of Technology and Design, Russia: A solution of the general Riesz-Radon problem for a Hausdorff space. Let T be a Hausdorff space with families G, F, C and B of open, closed, compact and Borel sets. Let RM_{0} be the set of all positive Radon measures μ : B → \overline{R}. Consider the algebra A ≡ \{∪(G_{i} ∩ F_{i} | i = 1, ..., n) | n ∈ N, G_{i} ∈ G, F_{i} ∈ F}, the set A_{c} ≡ {A ∈ A | ∃C ∈ C(A ⊂ C)}, the lattice linear spaceS_{c} of all functions f : T → R such that suppf ⊂ C ∈ C and for ϵ there is a finite cover (A_{i} ∈ A | i ∈ I) of T with ω(f, A_{i}) < ϵ, and the set (S_{c}^{△})_{+} of all positive linear functions ϕ : S_{c} → R such that ϕ is pointwise σ-continuous and for (A_{n} ∈ A_{c} | n ∈ N) ↓ and ϵ there are m and C ∈ C with C ⊂ ∩A_{n} and ϕ(χ(A_{m} \ C)) < ϵ.\underline{Theorem}. There is a bijection V : (S_{c}^{△})_{+} → RM_{0} such that ϕf = ∫ fdV ϕ for f ∈ S_{c} and ϕ ∈ (S_{c}^{△})_{+}. Part of the result is joint work with A. Mikhalev. Reference: V. K. Zakharov, and A. V. Mikhalev, An integral representation for Radon measures on an arbitrary Hausdorff space, Fund. i prikl. mat. 3(1997), 1135-1172 (in Russia).