PomeLemma 5.27.28.55862640C106 Let $\Omega \subset \mathbb C^ n$ be a domain. Suppose that $u_ j\in \mathrm{PSH}(\Omega )\cap L^\infty _{\mathrm{loc}}(\Omega )$ tend to zero in $L^1_{\mathrm{loc}}(\Omega )$ and satisfy $\sup _ j\| u_ j\| _{L^\infty (K)}<\infty $ for every compact set $K\subset \Omega $. Then $(\mathrm{dd}^{\mathrm{c}}u_ j)^ n(K)\to 0$ for every such $K$.
Lemma 5.27.2855862640C106
Proof. Fix concentric balls $B'\Subset B\Subset B_1\Subset \Omega $, and put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. If $\psi $ is a nonnegative smooth cutoff function supported in $B_1$ and equal to one near $\overline B$, integration by parts gives
\[ 0\leq \int _ B\mathrm{dd}^{\mathrm{c}}u_ j\wedge \beta ^{n-1} \leq \int _{B_1}u_ j\mathrm{dd}^{\mathrm{c}}\psi \wedge \beta ^{n-1} \leq C\| u_ j\| _{L^1(B_1)}\longrightarrow 0. \]
Choose $M>0$ with $|u_ j|\leq M$ on $B$ for all $j$, and apply the first estimate in (5.27.3) to $u_ j+M$, whose values lie in $[0,2M]$. It bounds the top-degree mass of $B'$ by $C(2M)^{n-1}$ times the trace integral over $B$, which tends to zero; for $n=1$ the trace estimate itself suffices. Covering $K$ by finitely many such balls proves the lemma. $\square$
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