PomeProposition 5.14.7.459E68F98F06 Let $(E_ i)_{i\in \mathbb {Z}_{>0}}$ be a sequence of pluripolar sets in $X$. Then
\[ E\coloneqq \bigcup _{i=1}^{\infty }E_ i \]
is also pluripolar.
Proposition 5.14.7459E68F98F06
Proof. The problem is local, so we may assume that $X\subseteq \mathbb {C}^ n$ is a bounded domain. In this case, by theorem 5.13.19 for each $i\in \mathbb {Z}_{>0}$ we can choose $\psi _ i\in \mathrm{PSH}(\mathbb {C}^ n)$ such that
\[ \psi _ i|_{E_ i}\equiv -\infty ,\quad \psi _ i|_ X\leq 0 \]
for all $i>0$. By proposition 5.14.1, we have $\psi _ i|_ X\in L^1(X)$ for all $i>0$. After rescaling, we may also assume that $\| \psi _ i\| _{L^1(X)}\leq 1$ for all $i>0$.
We then define
\[ \psi =\sum _{i=1}^{\infty }2^{-i} \psi _ i|_ X. \]
Then $\psi \in \mathrm{PSH}(X)$ according to proposition 5.14.2 and $\psi |_{E}=-\infty $. $\square$
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