Symmetric function kernels and sweeping of measures

TL;DR

This paper explores the theoretical aspects of sweeping positive Radon measures onto sets using symmetric function kernels, focusing on potential theory and capacity functional properties.

Contribution

It extends potential theoretic methods to sweeping measures onto quasiclosed sets via symmetric kernels and energy capacity functionals.

Findings

Finiteness of upper capacity ensures sweeping feasibility.

Develops a capacity-based framework for balayage.

Analyzes potentials with symmetric kernels in a general setting.

Abstract

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure on a locally compact (Hausdorff) space onto a closed, or more generally a quasiclosed set (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable positive symmetric function kernel. Following Choquet (1959) we consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions. The finiteness of the upper capacity of the potential restricted to the set in question is sufficient for the possibility of the sweeping.