Corollary 4.1.24098882E3CBEB

Proof. By theorem 4.1.19, it remains to show that a polar set has outer capacity zero. By proposition 4.1.11, $A$ is contained in the pole set of a function $E_ d*\nu $ with $\nu $ finite, positive and compactly supported. Let $N$ be a compact neighborhood of $\overline A$ in $B$. The set $P=N\cap \{ E_ d*\nu =-\infty \} $ is Borel, being the intersection of $N$ with the open sets $\{ E_ d*\nu <-n\} $, $n\geq 1$. Every compact subset of $P$ is polar and therefore has capacity zero by theorem 4.1.17. By theorem 4.1.23, $C_ B^*(P)=0$, and monotonicity gives $C_ B^*(A)=0$. For the kernel capacities, a compact set has capacity zero if and only if it carries no probability measure of finite energy (theorem 4.1.7), that is, no nonzero positive measure of finite energy; by theorem 4.1.17 this happens if and only if it is polar. $\square$