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Title of Journal: J Math Sci

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Abbravation: Journal of Mathematical Sciences

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Springer US

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DOI

10.1007/s10857-006-9010-z

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1573-8795

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Characterization of radon integrals as linear func

Authors: V K Zakharov A V Mikhalev T V Rodionov
Publish Date: 2012/07/26
Volume: 185, Issue: 2, Pages: 233-281
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Abstract

The problem of characterization of integrals as linear functionals is considered in this paper It has its origin in the wellknown result of F Riesz 1909 on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and is directly connected with the famous theorem of J Radon 1913 on integral representation of bounded linear functionals by Lebesgue integrals on a compact in ℝ n After the works of J Radon M Fréchet and F Hausdorff the problem of characterization of integrals as linear functionals has been concretized as the problem of extension of Radon’s theorem from ℝ n to more general topological spaces with Radon measures This problem turned out to be difficult and its solution has a long and abundant history Therefore it may be naturally called the Riesz–Radon–Fréchet problem of characterization of integrals The important stages of its solution are connected with such eminent mathematicians as S Banach 1937–38 S Saks 1937–38 S Kakutani 1941 P Halmos 1950 E Hewitt 1952 R E Edwards 1953 Yu V Prokhorov 1956 N Bourbaki 1969 H K¨onig 1995 V K Zakharov and A V Mikhalev 1997 et al Essential ideas and technical tools were worked out by A D Alexandrov 1940–43 M N Stone 1948–49 D H Fremlin 1974 et al The article is devoted to the modern stage of solving this problem connected with the works of the authors 1997–2009 The solution of the problem is presented in the form of the parametric theorems on characterization of integrals These theorems immediately imply characterization theorems of the abovementioned authors

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