Abstract:
After the fundamental work of Riesz, Radon and Hausdorff in the period 1909–1914, the following problem of general Radon representation emerged: for any Hausdorff space find the space of linear functionals that are integrally representable by Radon measures. In the early 1950s, a partial solution of this problem (the bijective version) for locally compact spaces was obtained by Halmos, Hewitt, Edwards, Bourbaki and others. For bounded Radon measures on a Tychonoff space, the problem of isomorphic Radon representation was solved in 1956 by Prokhorov.
In this paper we give a possible solution of the problem of general Radon representation. To do this, we use the family of metasemicontinuous functions with compact support and the class of thin functionals. We present bijective and isomorphic versions of the solution (Theorems 1 and 2 of § 2.5). To get the isomorphic version, we introduce the family of Radon bimeasures.
Citation:
V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space”, Izv. Math., 63:5 (1999), 881–921
\Bibitem{ZakMik99}
\by V.~K.~Zakharov, A.~V.~Mikhalev
\paper The problem of general Radon representation for an arbitrary Hausdorff space
\jour Izv. Math.
\yr 1999
\vol 63
\issue 5
\pages 881--921
\mathnet{http://mi.mathnet.ru/eng/im265}
\crossref{https://doi.org/10.1070/im1999v063n05ABEH000265}
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\zmath{https://zbmath.org/?q=an:0967.28012}
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This publication is cited in the following 10 articles:
Machsoudi S. Rejali A., “on the Dual of Certain Locally Convex Function Spaces”, Bull. Iran Math. Soc., 41:4 (2015), 1003–1017
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
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
Heinz König, Measure and Integration, 2012, 149
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
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Characterization of Radon integrals as linear functionals”, J. Math. Sci., 185:2 (2012), 233–281
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Characterization of general Radon integrals”, Dokl. Math., 82:1 (2010), 613
V. K. Zakharov, “The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures”, Proc. Steklov Inst. Math., 248 (2005), 101–110
V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space. II”, Izv. Math., 66:6 (2002), 1087–1101
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