Cegrell's fixed-measure convergence lemma states that a uniformly bounded sequence of plurisubharmonic functions converging in $L^1_{\mathrm{loc}}$ converges in $L^1(\nu )$ on compact subsets, for every finite non-pluripolar measure $\nu $. The usual formulation, which allows passing to a subsequence, gives the same conclusion: apply it to every subsequence after extracting a further subsequence converging almost everywhere for the smooth volume, and use the subsequence criterion for convergence. This is the classical fixed-measure argument used in [Section 11.1.2, GZ17], and we use this local lemma as a standard input.
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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