Corollary 5.14.8.E23F3D6154AB Let $(\varphi _ j)_{j\in \mathbb {Z}_{>0}}$ be a sequence in $\mathrm{PSH}(X)$ such that $\varphi _ j\xrightarrow {L^1_{\mathrm{loc}}}\varphi \in \mathrm{PSH}(X)$. Then
02A6B28E8E8F
5.14.9
\begin{equation} \varphi =\mathop{\vphantom {\operatorname {sup}^{*}}\inf }\limits _{j>0} \operatorname *{sup}_{k\geq j}{}\! {}^{*}\varphi _ k, \end{equation}
and the set
\[ \left\{ x\in X: \varphi (x)\neq \varlimsup _{j\to \infty }\varphi _ j(x)\right\} \]
is pluripolar.
Proof.
We may assume that $X$ is connected.
We first observe that $(\varphi _ j)_ j$ is locally uniformly bounded from above. This follows from [Exercise 1.20, GZ17].
For each $j\geq 1$, let
\[ \psi _ j=\operatorname *{sup}_{k\geq j}{}\! {}^{*}\varphi _ k. \]
Then $\psi _ j\in \mathrm{PSH}(X)$ by proposition 5.14.2. Moreover, $(\psi _ j)_ j$ is a decreasing sequence and $\psi _ j\geq \varphi _ j$ for all $j$. After considering a subsequence of $(\varphi _ j)_ j$ which converges to $\varphi $ almost everywhere, we get $\varphi \leq \psi \coloneqq \inf _ j \psi _ j$ almost everywhere. In particular, $\psi \in \mathrm{PSH}(X)$ by proposition 5.14.2.
On the other hand, by proposition 5.14.5, there exist pluripolar sets $Z_ j\subseteq X$ such that
\[ \psi _ j=\sup _{k\geq j}\varphi _ k \]
on $X\setminus Z_ j$. Let
\[ Z=\bigcup _{j=1}^{\infty } Z_ j. \]
Then $Z$ is a pluripolar set by proposition 5.14.7, and for any $x\in X\setminus Z$, we have $\psi (x)=\varlimsup _ j \varphi _ j(x)$. Moreover, $\psi =\varphi $ almost everywhere by [Theorem 1.46, GZ17], and hence everywhere by proposition 5.14.6. Hence we conclude (5.14.9). Moreover, for all $x\in X\setminus Z$,
\[ \varphi (x)=\varlimsup _{j\to \infty } \varphi _ j(x). \]
This completes the proof.
$\square$
Pome
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