On the theory of balayage on locally compact spaces

TL;DR

This paper advances the theory of balayage of Radon measures with finite energy on locally compact spaces, establishing new approximation and reduction results, and analyzing convergence properties for a broad class of kernels.

Contribution

It specifies the balayage theory for spaces with countable bases, reduces inner balayage to Borel sets, and characterizes balayage measures via symmetry relations.

Findings

Inner balayage reduces to Borel sets.

Existence of a $K_\sigma$-set $A_0$ with equal balayage measures.

Convergence analysis of swept measures and potentials.

Abstract

The paper deals with the theory of balayage of Radon measures $\mu$ of finite energy on a locally compact space $X$ with respect to a consistent kernel $\kappa$ satisfying the domination principle. Such theory is now specified for the case where the topology on $X$ has a countable base, while any $f\in C_0(X)$, a continuous function on $X$ of compact support, can be approximated in the inductive limit topology on the space $C_0(X)$ by potentials $\kappa\lambda:=\int\kappa(\cdot,y)\,d\lambda(y)$ of measures $\lambda$ of finite energy. In particular, we show that then the inner balayage can always be reduced to balayage to Borel sets. In more details, for arbitrary $A\subset X$, there exists a $K_\sigma$-set $A_0\subset A$ such that $\mu^A=\mu^{A_0}=\mu^{*A_0}$ for all $\mu$, $\mu^A$ and $\mu^{*A}$ denoting the inner and the outer balayage of $\mu$ to $A$, respectively. Furthermore, $\mu^A$ is now uniquely determined by the symmetry relation $\int\kappa\mu^A\,d\lambda=\int\kappa\lambda^A\,d\mu$, $\lambda$ ranging over a certain countable family of measures depending on $X$ and $\kappa$ only. As an application of these theorems, we analyze the convergence of inner and outer swept measures and their potentials. The results obtained do hold for many interesting kernels in classical and modern potential theory on $\mathbb R^n$, $n\geqslant2$.