Theorem 4.1.175B4ACD7E054D

Proof. Conditions (1)–(3) do not involve $B$, so for their equivalence we may enlarge $R$ and assume that $K\subset B$. We first show, for every ball $B=B(0,R)$ containing $K$, that $K$ is polar if and only if $C_ B(K)=0$. Since $u_ K=-G_ B\lambda _ K$ and $C_ B(K)=\lambda _ K(K)$ (proposition 4.1.16), we have $C_ B(K)=0$ if and only if $u_ K=0$. Suppose $u_ K=0$. Then $f_ K=0$ almost everywhere by theorem 3.4.23. Choose a point $x\in B$ with $f_ K(x)=0$ and admissible functions $v_ j$ with $v_ j(x)>-2^{-j}$. The partial sums $\sum _{j\leq N}v_ j$ are subharmonic and decrease with $N$, and their values at $x$ stay above $-1$. By theorem 3.4.9, the limit $\sum _ jv_ j$ is subharmonic on $B$ and not identically $-\infty $. It equals $-\infty $ on $K$, since every $v_ j$ is at most $-1$ there. Thus $K$ is polar.

Conversely, if $K$ is polar, proposition 4.1.11 gives a function $E_ d*\nu $, not identically $-\infty $, that equals $-\infty $ on $K$; subtracting a constant, we may assume it is negative on $B$. All its positive multiples are then admissible, so $f_ K=0$ wherever $E_ d*\nu >-\infty $, that is, almost everywhere. Hence $u_ K=0$ almost everywhere, and therefore everywhere by theorem 3.4.12.

Suppose that $K$ is not polar, so that $\lambda _ K\neq 0$. By parts (2) and (3) of proposition 4.1.16, $u_ K=E_ d*\lambda _ K+h$ with the harmonic function $h=-H_ B\bigl ((E_ d*\lambda _ K)|_{\partial B}\bigr )$ on $B$. Since $u_ K\geq -1$, it follows that $E_ d*\lambda _ K\geq -1-h>-\infty $ on $K$. Theorem 4.1.10 writes $\lambda _ K$ as a sum of measures with finite continuous potentials, and one of them, $\mu $, is nonzero; so (3) holds. A measure $\mu $ as in (3) has finite energy, because its potential $U^\mu $ is bounded on the compact support of $\mu $; so (3) implies (2).

Next, let $\mu $ be a finite positive measure with compact support such that $E_ d*\mu $ is finite and continuous, and let $\nu $ be any finite positive measure with compact support. We show that $\mu $ gives measure zero to the Borel set $\{ E_ d*\nu =-\infty \} $; by proposition 4.1.11, every bounded polar set lies in a set of this form. By the reciprocity law (4.1.2), $\int (E_ d*\nu )\, \mathrm d\mu =\int (E_ d*\mu )\, \mathrm d\nu $. The right side is finite because $E_ d*\mu $ is continuous, hence so is the left side, and therefore $E_ d*\nu >-\infty $ $\mu $-almost everywhere.

Now let $\mu $ be a finite positive measure with compact support and finite energy. Then $U^\mu <\infty $ $\mu $-almost everywhere, that is, $E_ d*\mu $ is finite $\mu $-almost everywhere, and theorem 4.1.10 writes $\mu $ as a sum of measures with finite continuous potentials. If $A$ is a bounded polar set and $\nu $ is chosen for $A$ as in proposition 4.1.11, the previous paragraph shows that each of these measures, and hence $\mu $, gives measure zero to the Borel set $\{ E_ d*\nu =-\infty \} $, which contains $A$. An arbitrary polar set is a countable union of bounded polar sets, so it is contained in a countable union of Borel sets of $\mu $-measure zero; this proves the final assertion. In particular, a nonzero measure with finite energy cannot be carried by a polar compact set, so (2) implies (1). $\square$