Theorem 5.47.2017D9EEFA42DF

Proof. Since $X$ has a countable basis, it is covered by countably many charts; in each chart, the coordinate balls with rational centers and radii and with closure in the chart domain form a countable basis. Let $\mathcal Q$ be the union of these countable families; it is a countable basis of $X$.

(1) Let $U=\bigcup _ iU_ i$; if $I$ is empty, take $J$ empty. We call a set basic if it has the form $Q\cap \{ \psi >0\} $, where $Q\in \mathcal Q$ and $\psi $ is a bounded plurisubharmonic function on a neighborhood of $\overline Q$. Every point $a\in U$ lies in a basic set contained in some $U_ i$: if $a\in U_ i$, then proposition 5.47.15(2) gives a ball $B$ and a bounded plurisubharmonic function $\psi $ on a neighborhood of $\overline B$ with $a\in B\cap \{ \psi >0\} \subset U_ i$, and it suffices to choose $Q\in \mathcal Q$ with $a\in Q\subset B$.

Fix $Q\in \mathcal Q$, and let $\Psi _ Q$ be the set of bounded plurisubharmonic functions $\psi $ on neighborhoods of $\overline Q$ for which the basic set $Q\cap \{ \psi >0\} $ is contained in some $U_ i$. If $\Psi _ Q$ is empty, we put $P_ Q=\varnothing $ and select nothing for this $Q$. Otherwise, we may assume that all members of $\Psi _ Q$ take values in $[-1,1]$ on $Q$, since replacing $\psi $ by $\psi /c$, where $c>0$ bounds $|\psi |$ on $Q$, does not change $Q\cap \{ \psi >0\} $.

Let $F_ Q$ be the pointwise supremum of the members of $\Psi _ Q$ on $Q$. By Choquet's lemma (lemma 3.4.22) there are $\psi _{Q,1},\psi _{Q,2},\ldots $ in $\Psi _ Q$ whose supremum $F_ Q'$ satisfies $(F_ Q')^*=F_ Q^*$. By corollary 5.47.11(1) and the last assertion of theorem 5.47.8, the set $P_ Q=\{ F_ Q<F_ Q^*\} \cup \{ F_ Q'<(F_ Q')^*\} $ is pluripolar in $Q$, and hence in $X$. If $z\in Q\setminus P_ Q$ and $\psi (z)>0$ for some $\psi \in \Psi _ Q$, then $F_ Q'(z)=F_ Q^*(z)=F_ Q(z)>0$, so $\psi _{Q,k}(z)>0$ for some $k$. Therefore

For each $Q$ and $k$ choose an index $i(Q,k)\in I$ with $Q\cap \{ \psi _{Q,k}>0\} \subset U_{i(Q,k)}$, and let $J$ be the countable set of these indices. Every point of $U$ lies in some basic set $Q\cap \{ \psi >0\} $ with $\psi \in \Psi _ Q$, so $U\setminus \bigcup _{j\in J}U_ j$ is contained in $\bigcup _ QP_ Q$. This countable union of pluripolar sets is pluripolar by theorem 5.47.8.

(2) Apply the construction of (1) to the family consisting of the single set $U$. It selects countably many sets $Q\cap \{ \psi _{Q,k}>0\} $, where $Q\in \mathcal Q$ is a coordinate ball and $\psi _{Q,k}$ restricts to a bounded plurisubharmonic function on $Q$; these sets are contained in $U$, and their union $V$ satisfies $U\setminus V\subset \bigcup _ QP_ Q$, which is pluripolar. Each selected set is Borel, because $\psi _{Q,k}$ is upper semicontinuous, so the countable union $V$ is Borel. Finally, let $C$ and $\mu $ be as in the statement. The set $(U\setminus V)\cap C$ is pluripolar in $C$, so it has outer $\mu $-measure zero and is a null set for the completion of $\mu $. Since $U\cap C=(V\cap C)\cup \bigl ((U\setminus V)\cap C\bigr )$ and $V\cap C$ is Borel, $U\cap C$ is measurable for the completion of $\mu $. $\square$