Lemma 5.44.2.983B3E6349BB Let $\nu $ be a finite non-pluripolar measure on $X$. If uniformly bounded quasi-plurisubharmonic functions $a_ j,a$ have a common smooth lower curvature bound and $a_ j\to a$ in $L^1(X)$, then
\[ \int _ X|a_ j-a|\, \mathrm{d}\nu \longrightarrow 0. \]
If instead $u_ j,u\in \mathrm{PSH}(X,\theta )$ have a common upper bound and $u_ j\to u$ in $L^1(X)$, then, for every $b>0$,
\[ \int _ X|\mathrm{e}^{bu_ j}-\mathrm{e}^{bu}|\, \mathrm{d}\nu \longrightarrow 0. \]
Proof.
For the first statement, add a fixed smooth strictly plurisubharmonic function on each of finitely many coordinate balls, so that the resulting functions are uniformly bounded and plurisubharmonic there. Apply the local lemma on smaller compact balls covering $X$ and sum the integrals; the added functions cancel in the differences.
For the second statement, choose $A>0$ with $A\omega \geq \theta $. If $u_ j\leq C$, the convex chain rule gives $\mathrm{dd}^{\mathrm{c}}\mathrm{e}^{bu_ j}\geq -bA\mathrm{e}^{bC}\omega $, also for unbounded $u_ j$ by bounded truncation; here the exponential is given the value zero at a pole. Thus the functions $\mathrm{e}^{bu_ j}$ are uniformly bounded and quasi-plurisubharmonic, with a common lower curvature bound. They converge in $L^1(\mathrm{d}V)$, because the exponential is Lipschitz with constant $b\mathrm{e}^{bC}$ on $(-\infty ,C]$. The first statement now applies.
$\square$
Pome
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