On Evans’ and Choquet’s Theorems for Polar Sets

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 P such that its Gμ is infinite set admits exactly if only Gδ-set. A known application Evans’ theorem yields the solutions generalized Dirichlet problem open sets by Perron-Wiener-Brelot method using harmonic upper lower functions. It shown that, an elementary “metric sweeping” measures without any can be obtained general satisfying local triangle property, property which amounts to being locally equivalent some negative power metric. The particular case, G(x,y) = |x − y|α−d ${\mathbbm {R}^{d}}$ , 2 < α d, solves long-standing problem.