Theorem 7.5.2C39FEC1AB985

Proof. Step 1: locally bounded $u$. Let $u$ be locally bounded, $K\subset D$ compact and $\varepsilon >0$; we find an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\varepsilon $ such that $u$ is continuous on $K\setminus G$. For each $x\in K$ choose $r_ x>0$ with $\overline{B(x,3r_ x)}\subset D$; finitely many balls $B(x_ i,r_{x_ i})$, $1\le i\le m$, cover $K$. Suppose that for each $i$ we find an open set $G_ i$ with $\operatorname {Cap}_ D(G_ i)<\varepsilon /m$ such that $u$ is continuous on $\overline{B(x_ i,r_{x_ i})}\setminus G_ i$. Then $G=\bigcup _ iG_ i$ satisfies $\operatorname {Cap}_ D(G)<\varepsilon $, and $u$ is continuous on $K\setminus G$, because the finitely many sets $(K\setminus G)\cap \overline{B(x_ i,r_{x_ i})}$ are closed in $K\setminus G$ and cover it, and $u$ is continuous on each of them. We may therefore assume that $K=\overline{B'}$, where $B'=B(x,r)$, $B=B(x,2r)$ and $B_1=B(x,3r)$ satisfy $\overline{B_1}\subset D$.

Subtracting a constant, we may assume that $-N\le u\le -1$ on $B_1$. Let $j>2/r$. Since $U=B(x,5r/2)$, a neighborhood of $\overline B$, lies at distance $r/2>1/j$ from $\mathbb C^ n\setminus B_1$, the regularization $u*\rho _{1/j}$ is smooth and plurisubharmonic on $U$ and takes values in $[-N,-1]$ there; moreover, $u*\rho _{1/j}$ decreases to $u$ as $j\to \infty $ (proposition 5.9.2). Put $\psi =|z-x|^2-4r^2$, choose $\delta >0$ with $\overline{B'}\subset \{ \psi <-\delta \} $, and let $g=\max \{ u,A\psi \} $ and $g_ j=\max \{ u*\rho _{1/j},A\psi \} $, with $A=N/\delta $, be the functions (7.3.2) on $U$. The $g_ j$ are continuous and plurisubharmonic on $U$ and decrease to $g$; moreover, $g=u$ on $\{ \psi <-\delta \} \supset \overline{B'}$, and $g_ j=g=A\psi $ on a collar of $\partial B$.

For $\delta '>0$ put $G_{j,\delta '}=\{ z\in B:g_ j(z)-g(z)>\delta '\} $; it is open, because $g_ j$ is continuous and $g$ is upper semicontinuous. By (7.4.6), applied with $h=g_ j$, $u=g$ and $\delta '$ in place of $\delta $,

The right-hand side tends to zero as $j\to \infty $ by dominated convergence: $(\mathrm{dd}^{\mathrm{c}}g)^ n$ has finite mass on $B$, since $g$ is bounded and plurisubharmonic near $\overline B$, and the functions $g_ j-g\ge 0$ are bounded on $B$ uniformly in $j$ and decrease to zero. Choose $j_1<j_2<\cdots $ such that $G_ l=G_{j_ l,2^{-l}}$ satisfies $\operatorname {Cap}_ D(G_ l)<\varepsilon 2^{-l}$, and let $G=\bigcup _ lG_ l$. Then $\operatorname {Cap}_ D(G)<\varepsilon $, and on $B\setminus G$ we have $0\le g_{j_ l}-g\le 2^{-l}$ for every $l$. Thus $g$ is a uniform limit of continuous functions on $B\setminus G$, hence continuous there; in particular $u=g$ is continuous on $\overline{B'}\setminus G$.

Step 2: arbitrary $u$. Let $K\subset D$ be compact and $\varepsilon >0$. By proposition 7.4.4, applied to $K$ and a compact neighborhood $L$ of $K$ in $D$, there is $s>0$ with $\operatorname {Cap}^*_ D(K\cap \{ u<-s\} )<\varepsilon /2$, and hence an open set $G_1\supset K\cap \{ u<-s\} $ with $\operatorname {Cap}_ D(G_1)<\varepsilon /2$. By Step 1, applied to the locally bounded plurisubharmonic function $\max \{ u,-s\} $, there is an open set $G_2$ with $\operatorname {Cap}_ D(G_2)<\varepsilon /2$ such that $\max \{ u,-s\} $ is continuous on $K\setminus G_2$. On $K\setminus (G_1\cup G_2)$ we have $u\ge -s$, hence $u=\max \{ u,-s\} $. Thus $G=G_1\cup G_2$ contains $K\cap \{ u=-\infty \} $, $\operatorname {Cap}_ D(G)<\varepsilon $, and $u$ is finite and continuous on $K\setminus G$.

Step 3: a single exceptional set. Choose compact sets $K_ l$ with $K_ l\subset \operatorname{Int}K_{l+1}$ and $\bigcup _ lK_ l=D$. By Step 2 there are open sets $G^{(l)}\supset K_ l\cap \{ u=-\infty \} $ with $\operatorname {Cap}_ D(G^{(l)})<\varepsilon 2^{-l}$ such that $u$ is finite and continuous on $K_ l\setminus G^{(l)}$. Put $G=\bigcup _ lG^{(l)}$. Then $\operatorname {Cap}_ D(G)<\varepsilon $, and $G$ contains the pole set of $u$. Every $x\in D\setminus G$ lies in some $\operatorname{Int}K_ l$, and on the neighborhood $\operatorname{Int}K_ l\setminus G$ of $x$ in $D\setminus G$ the function $u$ is a restriction of the continuous function $u|_{K_ l\setminus G^{(l)}}$. Hence $u|_{D\setminus G}$ is finite and continuous. $\square$