Theorem 5.10.2D32A2FB94F6B

Proof. Since $F^*$ is defined by a local formula and plurisubharmonicity is a local property, it suffices to prove that $F^*$ is plurisubharmonic on every connected coordinate neighborhood $V$, which we identify with a domain in $\mathbb {C}^ n$. By theorem 3.4.23, $F^*$ is subharmonic on $V$, and $F$ is Lebesgue measurable and equal to $F^*$ almost everywhere on $V$. By Choquet's lemma, lemma 3.4.22, there is a countable set $J_ V\subset I$ with $(\sup _{i\in J_ V}u_ i)^*=F^*$ on $V$.

We next show that $\mathrm{dd}^{\mathrm{c}}F^*\geq 0$ on $V$. Fix $i_0\in I$, enumerate $J_ V\cup \{ i_0\} $ as $i_0,i_1,i_2,\ldots $, with repetitions if this set is finite, and put $v_ j=\max \{ u_{i_0},\ldots ,u_{i_ j}\} $. The functions $v_ j$ are plurisubharmonic by proposition 5.9.5, increase with $j$, are locally uniformly bounded above, and dominate the locally integrable function $u_{i_0}$. Their limit $G=\sup _ jv_ j$ lies between $\sup _{i\in J_ V}u_ i$ and $F$, so $G^*=F^*$, and hence $G=F^*$ almost everywhere by theorem 3.4.23 applied to the countable family. On a compact set $K\subset V$ on which every $u_ i$ is bounded above by $M$, we have $0\leq v_ j-u_{i_0}\leq M-u_{i_0}$, so dominated convergence gives $v_ j\to F^*$ in $L^1(K)$. Therefore $\mathrm{dd}^{\mathrm{c}}F^*\geq 0$, as a limit of positive currents. Since $F^*$ is subharmonic and dominates the locally integrable function $u_{i_0}$, it is locally integrable and hence the canonical representative of its class, and theorem 5.9.1 shows that $F^*$ is plurisubharmonic on $V$.

If $X$ has a countable basis, cover $X$ by countably many connected coordinate neighborhoods $V_1,V_2,\ldots $ and put $J=\bigcup _ mJ_{V_ m}$. On each $V_ m$ we have $F^*=(\sup _{i\in J_{V_ m}}u_ i)^*\leq (\sup _{i\in J}u_ i)^*\leq F^*$, and since the regularization is local, $(\sup _{i\in J}u_ i)^*=F^*$ on $X$.

Finally, in each chart the functions $u_ i$ are subharmonic, so the second assertion of theorem 4.1.26 shows that $\{ F<F^*\} $ is polar in the corresponding open subset of $\mathbb R^{2n}$, that is, real polar. $\square$