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 Uspekhi Mat. Nauk, 2010, Volume 65, Issue 4(394), Pages 153–178 (Mi umn9362)

The Riesz–Radon–Fréchet problem of characterization of integrals

V. K. Zakharovab, A. V. Mikhalevb, T. V. Rodionova

a M. V. Lomonosov Moscow State University
b Centre for New Information Technologies, Moscow State University

Abstract: This paper is a survey of results on characterizing integrals as linear functionals. It starts from the familiar result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a closed interval, and is directly connected with Radon's famous theorem (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact subset of $\mathbb{R}^n$. After the works of Radon, Fréchet, and Hausdorff, the problem of characterizing integrals as linear functionals took the particular form of the problem of extending Radon's theorem from $\mathbb{R}^n$ to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and rich history. Therefore, it is natural to call it the Riesz–Radon–Fréchet problem of characterization of integrals. Important stages of its solution are associated with such eminent mathematicians as Banach (1937–1938), Saks (1937–1938), Kakutani (1941), Halmos (1950), Hewitt (1952), Edwards (1953), Prokhorov (1956), Bourbaki (1969), and others. Essential ideas and technical tools were developed by A. D. Alexandrov (1940–1943), Stone (1948–1949), Fremlin (1974), and others. Most of this paper is devoted to the contemporary stage of the solution of the problem, connected with papers of König (1995–2008), Zakharov and Mikhalev (1997–2009), and others. The general solution of the problem is presented in the form of a parametric theorem on characterization of integrals which directly implies the characterization theorems of the indicated authors.
Bibliography: 60 titles.

Keywords: Radon measure, regular measure, Radon integral, symmetrizable functions, tight functional, bimeasures.

DOI: https://doi.org/10.4213/rm9362

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English version:
Russian Mathematical Surveys, 2010, 65:4, 741–765

Bibliographic databases:

UDC: 517.987.1+517.982.3+517.518.1
MSC: Primary 28C05; Secondary 28A25, 28C15

Citation: V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The Riesz–Radon–Fréchet problem of characterization of integrals”, Uspekhi Mat. Nauk, 65:4(394) (2010), 153–178; Russian Math. Surveys, 65:4 (2010), 741–765

Citation in format AMSBIB
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This publication is cited in the following articles:
1. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Characterization of Radon integrals as linear functionals”, J. Math. Sci., 185:2 (2012), 233–281
2. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The characterization of integrals with respect to arbitrary Radon measures by the boundedness indices”, J. Math. Sci., 185:3 (2012), 417–429
3. D. A. Edwards, “On the representation of tight functionals as integrals”, Positivity, 17:4 (2013), 1101–1113
4. T. V. Rodionov, V. K. Zakharov, “A fine correlation between Baire and Borel functional hierarchies”, Acta Math. Hungar., 142:2 (2014), 384–402
5. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, J. Math. Sci., 213:2 (2016), 163–200
6. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Postclassical families of functions proper for descriptive and prescriptive spaces”, J. Math. Sci., 221:3 (2017), 360–383
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