On Evans’ and Choquet’s Theorems for Polar Sets

Abstract

By classical results of G.C. Evans and G. Choquet on “good” kernels G in potential theory, for every polar Kσ-set P, there exists a finite measure μ on P such that its potential Gμ is infinite on P, and a set P admits a finite measure μ on P such that Gμ is infinite exactly on P if and only if P is a polar Gδ-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. It is shown that, by an elementary “metric sweeping” of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = |x − y|α−d on \({\mathbbm {R}^{d}}\), 2 < α < d, solves a long-standing open problem.