Continuous functions on polar sets

Abstract:

Let $\Omega$ be a second countable Brelot harmonic space with a positive potential. If $K$ is a compact subset of $\Omega$ with more than one point, then $K$ is a polar set iff every positive continuous function on $K$ can be extended to a continuous potential on $\Omega = {{\mathbf {R}}^n}(n \geqslant 3)$. This is a generalization of the result proved by H. Wallin for the special case $\Omega = {{\mathbf {R}}^n}(n \geqslant 3)$ with Laplace harmonic space.