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 Rn. 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 Rn 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.
Citation:
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The Riesz–Radon–Fréchet problem of characterization of integrals”, Russian Math. Surveys, 65:4 (2010), 741–765
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\by V.~K.~Zakharov, A.~V.~Mikhalev, T.~V.~Rodionov
\paper The Riesz--Radon--Fr\'echet problem of characterization of integrals
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 4
\pages 741--765
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https://www.mathnet.ru/eng/rm9362
https://doi.org/10.1070/RM2010v065n04ABEH004693
https://www.mathnet.ru/eng/rm/v65/i4/p153
This publication is cited in the following 6 articles:
T. V. Rodionov, V. K. Zakharov, “A fine correlation between Baire and Borel functional hierarchies”, Acta Math. Hungar., 142:2 (2014), 384–402
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, J. Math. Sci., 213:2 (2016), 163–200
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
D. A. Edwards, “On the representation of tight functionals as integrals”, Positivity, 17:4 (2013), 1101–1113
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
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Characterization of Radon integrals as linear functionals”, J. Math. Sci., 185:2 (2012), 233–281
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