Characterization of integrals with respect to arbitrary radon measures by the boundedness indices

Abstract

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 \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, 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 \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich 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 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.