Corollary 7.6.2D865A8538D7D

Proof. By theorem 5.10.2, $u=F^*$ is plurisubharmonic, and there is a countable subfamily $\{ w_1,w_2,\ldots \} \subset \mathcal F$ with $(\sup _ kw_ k)^*=u$. The functions $v_ j=\max \{ w_1,\ldots ,w_ j\} $ are plurisubharmonic (proposition 5.9.5) and increase to $G=\sup _ kw_ k$, and $G=G^*=u$ almost everywhere by theorem 5.10.2, applied to the countable family. Fix an integer $m\ge 1$. The functions $\max \{ v_ j,-m\} $ are locally bounded and plurisubharmonic on $D$, and they increase almost everywhere to the locally bounded plurisubharmonic function $\max \{ u,-m\} $, so by theorem 7.6.1 they converge to it locally in capacity. For rational $\delta >0$, the set where $\sup _ j\max \{ v_ j,-m\} <\max \{ u,-m\} -\delta $ is contained in $\{ \max \{ v_ k,-m\} <\max \{ u,-m\} -\delta \} $ for every $k$, and the outer capacities of the intersections of the latter sets with a compact set tend to zero as $k\to \infty $. Taking the union over rational $\delta $, countable subadditivity shows that the set

meets every compact set in a set of outer capacity zero.

If $\sup _ jv_ j(x)<u(x)$, then $u(x)>-\infty $, so $u(x)>-m$ for some integer $m$, and then $x\in E_ m$. Hence $\{ \sup _ jv_ j<u\} \subset \bigcup _ mE_ m$ also meets every compact set in a set of outer capacity zero. Finally, $\sup _ jv_ j\le F\le u$, so $\{ F<F^*\} \subset \{ \sup _ jv_ j<u\} $. (Since outer capacity is defined for all sets, we need not know whether $\{ F<F^*\} $ is Borel.) $\square$