Characterization of radon integrals as linear functionals

Abstract

The problem of characterization of integrals as linear functionals is considered in this paper. It has its origin in the well-known result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann–Stieltjes 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 ℝn. After the works of J. Radon, M. Frechet, 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 ℝn to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and abundant history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution 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. K¨onig (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 the above-mentioned authors.