Proposition 7.4.4815E5BBFC34A

Proof. Choose an open neighborhood $U$ of $K$ with compact closure in $\operatorname{Int}L$. For every competitor $v$, theorem 7.4.1, applied to $\overline U\subset \operatorname{Int}L$ with $q=n$ and $u_1=\cdots =u_ n=v$, gives

Since $V$ is upper semicontinuous, $U\cap \{ V<-s\} $ is an open set containing $K\cap \{ V<-s\} $; taking the supremum over $v$ proves the estimate. Since $V$ is locally integrable (theorem 5.9.1), $\| V\| _{L^1(L)}<\infty $, and letting $s\to \infty $ gives $\operatorname {Cap}^*_ D(K\cap \{ V=-\infty \} )=0$.

Now let $E$ be as in the statement. Let $x\in D$, let $W$ and $V_ W$ be as in the hypothesis, and choose $r>0$ with $\overline{B(x,3r)}\subset W$; put $B_ x=B(x,3r)$, $K_ x=\overline{B(x,r)}$ and $L_ x=\overline{B(x,2r)}$. Then $E\cap B_ x\subset \{ V_ W=-\infty \} $, and the restriction of $V_ W$ to $B_ x$ is plurisubharmonic. The estimate just proved, applied in $B_ x$ to this restriction and to $K_ x\subset \operatorname{Int}L_ x$, gives $\operatorname {Cap}^*_{B_ x}(E\cap K_ x)=0$; since every open set in $B_ x$ is open in $D$, the first inequality of proposition 7.4.3(4) gives $\operatorname {Cap}^*_ D(E\cap K_ x)=0$. A compact set $K\subset D$ is covered by finitely many interiors $\operatorname{Int}K_ x$, and subadditivity gives $\operatorname {Cap}^*_ D(E\cap K)=0$. Finally, by theorem 7.4.1, applied on $B_ x$ to $K_ x\subset \operatorname{Int}L_ x$ with $q=n$, the function $V_ W$ is integrable on $K_ x$ for every measure $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$ as in the statement, so the Borel set $K_ x\cap \{ V_ W=-\infty \} $ has measure zero for all these measures. Since $D$ has a countable basis, countably many sets $\operatorname{Int}K_ x$ cover $D$, and the union of the corresponding sets $K_ x\cap \{ V_ W=-\infty \} $ is a Borel set with the required property. $\square$