Proposition 7.11.221036F9F7DDDA

Proof. (1) Let $h$ be a competitor for $\operatorname {Cap}_ B$, that is, $h\in \mathrm{PSH}(B)$ and $0\leq h\leq 1$. The set $\{ \varphi <-T\} $ is open, because $\varphi $ is upper semicontinuous, and it is relatively compact in $B$, because $\varphi $ has zero boundary values. Hence $\nu =\mathbf1_{\{ \varphi <-T\} }(\mathrm{dd}^{\mathrm{c}}h)^ n$ is a finite measure, and it is dominated by $(\mathrm{dd}^{\mathrm{c}}h)^ n$. The tail estimate (7.11.3) of theorem 7.11.2, applied to $\nu $ and $w=h$, involves the constant $\kappa _{n,p}=2^{n+p}(n+1)!M^ n$, where $M\leq 1$ is the oscillation of $h$ on $B$, but not the mass of $\nu $; it gives

Taking the supremum over $h$ proves (1).

(2) Choose an open set $U$ and compact sets $L,L'\subset B$ with $K\subset U\Subset \operatorname{Int}L$ and $L\subset \operatorname{Int}L'$. By the Chern–Levine–Nirenberg inequality (theorem 7.2.5, with the base current $1$ and the pair $L\subset \operatorname{Int}L'$), there is a constant $C_ L$ with $\int _ L(\mathrm{dd}^{\mathrm{c}}h)^ n\leq C_ L$ for every competitor $h$, since $\| h\| _{L^\infty (L')}\leq 1$. The finite measure $\mathbf1_ L(\mathrm{dd}^{\mathrm{c}}h)^ n$ is dominated by $(\mathrm{dd}^{\mathrm{c}}h)^ n$ and has mass at most $C_ L$, so the constant $A_ p$ of theorem 7.11.2 for this measure and $w=h$ is at most

Hence the moment estimate (7.11.4), applied to $\varphi _ j$, gives

for all $j$ and all competitors $h$.

We pass this bound to the limit $u$. Since the $\varphi _ j$ are nonpositive, $u\leq 0$ almost everywhere, and hence everywhere by the submean inequality. Fix $S>0$. The truncations $\max \{ \varphi _ j,-S\} $ take values in $[-S,0]$ and converge to the plurisubharmonic function $\max \{ u,-S\} $ in $L^1_{\mathrm{loc}}(B)$, since $t\mapsto \max \{ t,-S\} $ is $1$-Lipschitz. Since $\mathbf1_ L(\mathrm{dd}^{\mathrm{c}}h)^ n\leq (\mathrm{dd}^{\mathrm{c}}h)^ n$, lemma 7.11.9 shows that they also converge in $L^1(\mathbf1_ L(\mathrm{dd}^{\mathrm{c}}h)^ n)$. As $t\mapsto t^ p$ is Lipschitz on $[0,S]$ and $(-\max \{ \varphi _ j,-S\} )^ p\leq (-\varphi _ j)^ p$, it follows that

Letting $S\to \infty $, monotone convergence gives the same bound for $\int _ L(-u)^ p(\mathrm{dd}^{\mathrm{c}}h)^ n$. The set $U\cap \{ u<-T\} $ is open, because $u$ is upper semicontinuous, and it contains $K\cap \{ u<-T\} $. By Chebyshev's inequality its $(\mathrm{dd}^{\mathrm{c}}h)^ n$-measure is at most $C_{K,p}E_ p^{p/(n+p)}T^{-p}$. Taking the supremum over $h$ bounds $\operatorname {Cap}_ B(U\cap \{ u<-T\} )$, and hence the outer capacity in (7.11.23). Finally, $K\cap \{ u=-\infty \} $ is contained in $K\cap \{ u<-T\} $ for every $T$, so letting $T\to \infty $ shows that it has zero outer capacity. $\square$