Continuous functions on polar sets

Abstract

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