FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1997, VOLUME 3, NUMBER 4, PAGES 1135-1172

A. V. Mikhalev

V. K. Zakharov

Abstract

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After the fundamental articles of Riesz, Radon and Hausdorf in
1909--1914 the problem of \emph{general Radon representation}
became to be actual: \emph{for Hausdorf topological spaces to find
a class of linear functionals isomorphic to the space
of integrally represented Radon measures}. Up to the beginning
of 50-s a bijective solution of Radon representation for
locally compact spaces was obtained by Halmos, Hewitt, Edwards,
Bourbaki etc. For bounded Radon measures on a Tychonof space the problem
of bijective representation was solved in 1956 by Yu. V. Prohorov.
```

In 1975--1976 Topsoe and Pollard made an important step into
consideration of the problem for an arbitrary Hausdorf topological
space. On this way K\"onig in 1995--1997 has got a bijective
version of Radon representation for isotone and positively-linear
functionals on the cone of positive upper semicontinuous functions
with compact support.

In 1996--1997 the authors have got bijective and isomorphic
versions of the general Radon representation.

In this paper one of possible solution of the general Radon
representation is exposed. For this reason the family of
\emph{metasemicontinuous functions with compact support} and the class
of \emph{thin functionals} are under consideration. Bijective and
isomorphic versions of a solution (theorems 1 and 2 (II.5))
are given. To get an isomorphic version the family of Radon
bimeasures is introduced.

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