Balayage of measures on a locally compact space

DOI10.1007/S10476-022-0122-1arXiv2010.07199MaRDI QIDQ6351304

N. Zorii

Publication date: 14 October 2020

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 $AsubsetX$, generalizing Cartan's theory of Newtonian inner balayage on $mathbb{R}^n$, $ngeqslant3$, 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 $K_{s}igma$, 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 $mathbb{R}^n$, $ngeqslant2$, which is important in applications.

Mathematics Subject Classification ID

Potentials and capacities on other spaces (31C15)

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