FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 8, PAGES 87-161

**Characterization of Radon integrals as linear functionals**

V. K. Zakharov

A. V. Mikhalev

T. V. Rodionov

Abstract

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The problem of characterization of integrals as linear functionals is
considered in the paper.
It takes the origin in the well-known result of F. Riesz (1909)
on integral representation of bounded linear functionals by
Riemann--Stiltjes integrals on a segment and is directly
connected with the famous theorem of J. Radon (1913) on integral
representation of bounded linear functionals by Lebesgue integrals on
a compact in $$**R**^{n}.
After works of J. Radon, M. Fréchet, and
F. Hausdorff, the problem of characterization of integrals as
linear functionals has been concretized as the problem of extension of
Radon's theorem from $$**R**^{n} to
more general topological spaces with Radon measures.
This problem turned out difficult, and its solution has a long
and abundant history.
Therefore, it may be naturally called the Riesz--Radon--Fr'echet
problem of characterization of integrals.
The important stages of its solving are connected with such eminent
mathematicians as S. Banach (1937--38), S. Saks (1937--38),
S. Kakutani (1941), P. Halmos (1950), E. Hewitt (1952),
R. E. Edwards (1953), Yu. V. Prokhorov (1956),
N. Bourbaki (1969), H. König (1995),
V. K. Zakharov and A. V. Mikhalev (1997), et al.
Essential ideas and technical tools were worked out by
A. D. Alexandrov (1940--43), M. N. Stone
(1948--49), D. H. Fremlin (1974), et al.
The article is devoted to the modern stage of solving this problem
connected with the works of the authors (1997--2009).
The solution of the problem is presented in the form of the parametric
theorems on characterization of integrals.
These theorems immediately imply characterization theorems of
above-mentioned authors.

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Last modified: December 5, 2011