Theorem 4.1.26E54414B74C32

Proof. (1) Let $N\subset \Omega $ be compact, and take $R$ so large that $N\subset B$. The set $E=\{ x\in N:\varlimsup _ jv_ j(x)<v(x)\} $ is the union, over rational numbers $a<b$ and integers $J\geq 1$, of the Borel sets

so it is Borel. Suppose some compact subset $S$ of $E$ were not polar. By theorem 4.1.17, $S$ would carry a nonzero positive measure $\sigma $ with $E_ d*\sigma $ finite and continuous, and lemma 4.1.25 would give $\varlimsup _ jv_ j=v$ $\sigma $-almost everywhere on $S$, which is impossible. Hence every compact subset of $E$ is polar and has capacity zero. By theorem 4.1.23 and corollary 4.1.24, $E$ is polar. Since countably many compact sets $N$ cover $\Omega $, proposition 4.1.11 completes the proof.

(2) If the family is empty, or if all its members are identically $-\infty $ on a component of $\Omega $, then $F=F^*=-\infty $ there. Let $\Omega '$ be a component on which some member $u_{t_0}$ is not identically $-\infty $; by proposition 3.4.8, $u_{t_0}$ is locally integrable on $\Omega '$. By lemma 3.4.22, applied on $\Omega '$, there are members $u_{t_1},u_{t_2},\ldots $ of the family with $(\sup _{k\geq 1}u_{t_ k})^*=F^*$ on $\Omega '$. Put $w_ j=\max \{ u_{t_0},\ldots ,u_{t_ j}\} $ and $w=\sup _ jw_ j$. Since $\sup _{k\geq 1}u_{t_ k}\leq w\leq F$, we have $w^*=F^*$ on $\Omega '$, and theorem 3.4.23, applied to the family $(w_ j)$, shows that $F^*$ is subharmonic on $\Omega '$ and that $w=F^*$ almost everywhere on $\Omega '$.

We show next that $w_ j\to F^*$ in $L^1_{\mathrm{loc}}(\Omega ')$. As $u_{t_0}\leq w_ j\leq F^*$, the functions $w_ j$ and $F^*$ are locally integrable on $\Omega '$. If $Q\subset \Omega '$ is compact and $M$ is a common upper bound of the $w_ j$ on $Q$, then $0\leq M-w_ j\leq M-u_{t_0}$ on $Q$ and $M-w_ j$ decreases to $M-w$, so dominated convergence gives $w_ j\to w=F^*$ in $L^1(Q)$. Since $\varlimsup _ jw_ j=w$, part (1) shows that $\{ w<F^*\} \cap \Omega '$ is polar, and $\{ F<F^*\} \cap \Omega '$ is contained in it because $w\leq F$. The components of $\Omega $ are pairwise disjoint nonempty open sets, hence of positive Lebesgue measure, so there are at most countably many of them (lemma 2.2.5), and the union over them is polar by proposition 4.1.11. $\square$