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

This article is cited in 6 scientific papers (total in 6 papers)

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.


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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
Received: 25.02.2010

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
\by V.~K.~Zakharov, A.~V.~Mikhalev, T.~V.~Rodionov
\paper The Riesz--Radon--Fr\'echet problem of characterization of integrals
\jour Uspekhi Mat. Nauk
\yr 2010
\vol 65
\issue 4(394)
\pages 153--178
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 4
\pages 741--765

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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  mathnet  crossref  mathscinet
    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  mathnet  crossref
    3. D. A. Edwards, “On the representation of tight functionals as integrals”, Positivity, 17:4 (2013), 1101–1113  crossref  mathscinet  zmath  isi  scopus
    4. T. V. Rodionov, V. K. Zakharov, “A fine correlation between Baire and Borel functional hierarchies”, Acta Math. Hungar., 142:2 (2014), 384–402  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref  mathscinet
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