Balayage of Radon measures of infinite energy on locally compact spaces

TL;DR

This paper develops a comprehensive theory of balayage for Radon measures with infinite energy on locally compact spaces, extending classical potential theory to more general measures and sets.

Contribution

It introduces new methods for balayage of Radon measures without finite energy, generalizing existing theories and providing a framework applicable to various kernels in potential theory.

Findings

Established existence and uniqueness of balayage measures.

Provided alternative characterizations of balayage.

Extended balayage theory to measures of infinite energy.

Abstract

For suitable kernels on a locally compact space $X$, we develop a theory of inner balayage of quite general Radon measures $\omega$ (not necessarily of finite energy) to arbitrary $A\subset X$. In the case where $A$ is Borel, this theory provides, as a by-product, a theory of outer balayage. We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established generalizes substantially the existing ones, pertaining either to $\omega$ of finite energy, or to some particular $A$ (e.g. quasiclosed). This work covers many interesting kernels in classical and modern potential theory, which looks promising for possible applications.