Balayage of measures on a locally compact space

TL;DR

This paper develops a comprehensive theory of inner balayage of positive Radon measures on locally compact spaces, generalizing classical Newtonian balayage and establishing new results on measure and potential convergence.

Contribution

It generalizes Cartan's Newtonian balayage to broader kernels on locally compact spaces and extends Fuglede's outer balayage results to arbitrary Borel sets.

Findings

Established a unified framework for inner and outer balayage.

Provided formulas for total mass evaluation of balayage measures.

Proved convergence theorems for balayage measures and potentials.

Abstract

We develop a theory of inner balayage of a positive Radon measure $\mu$ of finite energy on a locally compact space $X$ to arbitrary $A\subset X$, generalizing Cartan's theory of Newtonian inner balayage on $\mathbb R^n$, $n\geqslant3$, to a suitable function kernel on $X$. As an application of the theory thereby established, we show that if the space $X$ is perfectly normal and of class $K_\sigma$, then a recent result by Bent Fuglede (Anal. Math., 2016) on outer balayage of $\mu$ to quasiclosed $A$ remains valid for arbitrary Borel $A$. We give in particular various alternative definitions of inner (outer) balayage, provide a formula for evaluation of its total mass, and prove convergence theorems for inner (outer) swept measures and their potentials. The results obtained do hold (and are new in part) for most classical kernels on $\mathbb R^n$, $n\geqslant2$, which is important in applications.