Paper detail

On Evans' and Choquet's theorems on polar sets

By classical results of G.C. Evans and G. Choquet on &#34;good kernels $G$ in potential theory&#34;, for every polar $K_σ$-set $P$, there exists a finite measure $μ$ on $P$ such that $Gμ=\infty$ on $P$, and a set $P$ admits a finite measure $μ$ on $P$ such that $\{Gμ=\infty\}=P$ if and only if $P$ is a polar $G_δ$-set. A known application of Evans&#39; 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 &#34;metric&#34; considerations and without using any potential theory, such results can be obtained for general kernels $G$ satisfying a local triangle property. The particular case, $G(x,y)=|x-y|^{α-d}$ on $R^d$, $2<α<d$, solves a long-standing open problem.