FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 107-126

**The characterization of integrals with respect to arbitrary Radon
measures by the boundedness indices**

V. K. Zakharov

A. V. Mikhalev

T. V. Rodionov

Abstract

View as HTML
View as gif image

The problem of characterization of integrals as linear functionals is
considered in the paper.
It starts from the familiar results of F. Riesz (1909) and
J. Radon (1913) on integral representation of bounded linear
functionals by Riemann--Stieltjes integrals on a segment and by
Lebesgue integrals on a compact in $$**R**^{n},
respectively.
After works of J. Radon, M. Fréchet, and
F. Hausdorff the problem of characterization of integrals as
linear functionals took the particular form of the problem of
extension of Radon's theorem from $$**R**^{n} to
more general topological spaces with Radon measures.
This problem has 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
mathematicians as S. Banach, S. Saks, S. Kakutani,
P. Halmos, E. Hewitt, R. E. Edwards,
N. Bourbaki, V. K. Zakharov, A. V. Mikhalev,
et al.
In this paper, the Riesz--Radon--Fr'echet problem is solved for the
general case of arbitrary Radon measures on Hausdorff spaces.
The solution is given in the form of a general parametric theorem
in terms of a new notion of the *boundedness index of
a functional*.
The theorem implies as particular cases well-known results of the
indicated authors characterizing Radon integrals for various classes
of Radon measures and topological spaces.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k111/k11106h.htm

Last modified: January 31, 2012