On Evans' and Choquet's theorems on polar sets

Abstract

By classical results of G.C. Evans and G. Choquet on "good kernels $G$ in potential theory", for every polar $K_\sigma$-set $P$, there exists a finite measure $\mu$ on $P$ such that $G\mu=\infty$ on $P$, and a set $P$ admits a finite measure $\mu$ on $P$ such that $\{G\mu=\infty\}=P$ if and only if $P$ is a polar $G_\delta$-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. In this note it is shown that, by elementary "metric" considerations and without using any potential theory, such results can be obtained for general kernels $G$ satisfying a local triangle property. The particular case, $G(x,y)=|x-y|^{\alpha-d}$ on $R^d$, $2<\alpha<d$, solves a long-standing open problem.