Proposition 5.47.71D1F65765AD9

Proof. Choose compact sets $K_ j$ with $K_ j\subset \operatorname{Int}K_{j+1}$ and $\bigcup _ jK_ j=G$. Since the $K_ j$ increase, the functions $H_{K_ j}$ decrease. They take values in $[-1,0]$, so their limit $H$ is plurisubharmonic by theorem 5.10.1. Since $h_{K_ j}=-1$ on $K_ j$ and the upper regularization at a point only involves the values of $h_{K_ j}$ near that point, we have $H_{K_ j}=-1$ on $\operatorname{Int}K_ j$. Every point of $G$ lies in $\operatorname{Int}K_ j$ for all large $j$, so $H=-1$ on $G$. Thus $H$ is a competitor for $G$, and $H\le h_ G$. Conversely, every competitor for $G$ is a competitor for each $K_ j$, hence bounded above by every $H_{K_ j}$ and thus by $H$; so $h_ G\le H$. We have shown $h_ G=H$. As $H$ is upper semicontinuous, $H_ G=h_ G^*=H$; in particular $H_ G=-1$ on $G$. The boundary behavior follows from (5.47.3).

By theorem 7.3.7, $(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n\to (\mathrm{dd}^{\mathrm{c}}H_ G)^ n$ weakly. By proposition 5.47.6, these measures are carried by $K_ j\subset \overline G$. Hence the limit measure vanishes on the open set $B\setminus \overline G$, that is, it is carried by the compact set $\overline G$, and testing with a function in $C_ c(B)$ equal to one on $\overline G$ shows that the total masses converge. Since the Borel sets $K_ j$ increase to $G$, proposition 7.4.3(2) gives $\operatorname {Cap}_ B(K_ j)\to \operatorname {Cap}_ B(G)$. Together with the capacity formula of proposition 5.47.6 for the $K_ j$, this yields $\operatorname {Cap}_ B(G)=\lim _ j\int _ B(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n=\int _ B(\mathrm{dd}^{\mathrm{c}}H_ G)^ n$. $\square$