PomeTheorem 5.10.4.C2DB8DFE9E2C Let $\Omega \subset \mathbb C^ n$ be a domain, $n\geq 1$. Every sequence in $\mathrm{PSH}(\Omega )$ that is locally uniformly bounded above has a subsequence that either converges locally uniformly to $-\infty $ or converges in $L^1_{\mathrm{loc}}(\Omega )$ to a plurisubharmonic function. If a sequence $(u_ j)$ in $\mathrm{PSH}(\Omega )$ converges in $L^1_{\mathrm{loc}}(\Omega )$, then its limit has a plurisubharmonic representative $u$, and the following holds: for every compact set $K\subset \Omega $ and every lower semicontinuous function $f\colon K\to \mathbb R$ with $u<f$ on $K$, for instance every real continuous function on $K$, one has $u_ j<f$ on $K$ for all sufficiently large $j$. In particular $\varlimsup _ j u_ j(x)\le u(x)$ for every $x\in \Omega $.
Theorem 5.10.4C2DB8DFE9E2C
Proof. Plurisubharmonic functions are subharmonic by theorem 5.9.1. Hence, by the compactness theorem (theorem 3.4.19), a sequence as in the first assertion either tends to $-\infty $ locally uniformly, or has a subsequence converging in $L^1_{\mathrm{loc}}(\Omega )$ to a locally integrable subharmonic function $v$. In the second case $\mathrm{dd}^{\mathrm{c}}v\geq 0$, as a distributional limit of positive currents, and $v$ is the canonical representative of its class; so $v$ is plurisubharmonic by theorem 5.9.1.
Now let $(u_ j)$ converge in $L^1_{\mathrm{loc}}(\Omega )$ to $T$. As above, $\mathrm{dd}^{\mathrm{c}}T\geq 0$, so $T$ has a plurisubharmonic representative $u$, namely its canonical subharmonic representative. The $u_ j$ are locally integrable subharmonic functions converging in $L^1_{\mathrm{loc}}(\Omega )$, and $u$ is the canonical representative of their limit, so the remaining assertions are Hartogs' lemma for subharmonic functions, theorem 3.4.20, which requires no upper bound on the sequence; the last one is its first assertion with the constant sequence $x_ j=x$. $\square$
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