Proposition 4.1.1164A94A9EE9E3

Proof. Fix a bounded open set $W\supset \overline A$; we construct $\nu $ with support in $\overline W$. Let $a\in A$. By definition 3.4.26, after shrinking the neighborhood as explained above, there are an open ball $V\subset W$ centered at $a$ and a nontrivial function $\psi \in \mathrm{SH}(V)$ with $A\cap V\subset \{ \psi =-\infty \} $; by proposition 3.4.8, $\psi $ is locally integrable. Choose an open ball $B_ a$ centered at $a$ with $\overline{B_ a}\subset V$, and a cutoff $\eta \in C_ c^\infty (V)$ with $0\leq \eta \leq 1$ and $\eta =1$ near $\overline{B_ a}$. By the Riesz decomposition (theorem 3.4.14), $\psi =E_ d*(\eta \mu _\psi )+h$ on $B_ a$, where $\mu _\psi $ is the Riesz measure of $\psi $ and $h$ is harmonic, hence finite; therefore $E_ d*(\eta \mu _\psi )=-\infty $ on $A\cap B_ a$. The measure $\eta \mu _\psi $ is finite, positive and supported in $\overline W$. Since $\mathbb R^ d$ has a countable base, countably many of the balls $B_ a$ cover $A$. Let $\nu _ j$ be the corresponding measures.

Choose $c_ j>0$ with $\sum _ jc_ j\nu _ j(\mathbb R^ d)<\infty $, and put $\nu =\sum _ jc_ j\nu _ j$, a finite positive measure supported in $\overline W$. Let $S\subset \mathbb R^ d$ be compact and let $M$ be an upper bound of $E_ d$ on the compact set $S-\overline W$. For $x\in S$, Tonelli's theorem applied to the nonnegative functions $M-E_ d(x-\cdot )$ gives

Thus $E_ d*\nu (x)=-\infty $ whenever $E_ d*\nu _ j(x)=-\infty $ for some $j$; in particular $E_ d*\nu =-\infty $ on $A$. By proposition 3.4.11, $E_ d*\nu $ is subharmonic and locally integrable, so it is not identically $-\infty $.

Now let $A=\bigcup _ kA_ k$ with polar sets $A_ k$, let $x\in \mathbb R^ d$, and let $B'$ be an open ball centered at $x$. Apply the construction above to the bounded polar sets $A_ k\cap B'$, with one and the same bounded open set $W\supset \overline{B'}$. It gives measures $\nu ^{(k)}$ supported in $\overline W$ with $E_ d*\nu ^{(k)}=-\infty $ on $A_ k\cap B'$. Choosing $c_ k>0$ with $\sum _ kc_ k\nu ^{(k)}(\mathbb R^ d)<\infty $, the Tonelli argument above shows that $\nu =\sum _ kc_ k\nu ^{(k)}$ satisfies $E_ d*\nu =-\infty $ on $A\cap B'$. Since $E_ d*\nu $ is subharmonic on $B'$ and not identically $-\infty $ there, $A$ is polar. $\square$