Lemma 16.11.21BA71B27F698

Proof. (1) By lemma 5.4.2 every point of $X$ has a neighborhood on which $\theta '=\mathrm{dd}^{\mathrm{c}}h$ for a smooth function $h$, so $X$ is covered by finitely many coordinate balls $D_ i$ on which $\theta '=\mathrm{dd}^{\mathrm{c}}h_ i$ with $h_ i$ smooth near $\overline{D_ i}$. On $D_ i$ the function $u+h_ i$ is plurisubharmonic by lemma 6.14.4(2), hence quasi-continuous by theorem 7.5.2, and so is $u=(u+h_ i)-h_ i$, because $h_ i$ is continuous and real. By lemma 16.11.1(3), $u$ is quasi-continuous on every chart.

(2) Let $f_1,\ldots ,f_ k$ be quasi-continuous, let $K$ be a compact subset of a chart $D$, and let $\varepsilon >0$. Choose open sets $G_ i\subset D$ with $\operatorname {Cap}_ D(G_ i)<\varepsilon /k$ such that $f_ i$ is continuous on $K\setminus G_ i$. By subadditivity, $G=G_1\cup \cdots \cup G_ k$ has capacity less than $\varepsilon $, and all the $f_ i$, hence also their maximum and minimum, are continuous on $K\setminus G$.

(3) Let $D$ be a chart, $K\subset D$ compact and $\varepsilon >0$. The set $\{ f\ne g\} $ is pluripolar, so by lemma 16.11.1(2) its intersection with $D$ is contained in an open set $G'\subset D$ with $\operatorname {Cap}_ D(G')<\varepsilon /2$. If $G\subset D$ is an open set with $\operatorname {Cap}_ D(G)<\varepsilon /2$ such that $f$ is continuous on $K\setminus G$, then $\operatorname {Cap}_ D(G\cup G')<\varepsilon $, and $g=f$ is continuous on $K\setminus (G\cup G')$. $\square$