If the potentials $\psi _ j$ converge in capacity, the integration error in proposition 15.37.17 tends to zero, and the common lower envelope controls the measures of deep sublevel sets. The dominating measure may also contain a fixed finite non-pluripolar part.
Theorem 15.37.26.C73D2354DDB2 Let $u_ j\in \mathcal E_\varphi $, and suppose that, for a fixed $h\in \mathcal E_\varphi $ and a constant $C$,
\[ h\leq u_ j\leq \varphi +C,\qquad u_ j\longrightarrow u\in \mathrm{PSH}(X,\theta )\quad \hbox{in }L^1(X). \]
Fix $A\geq 0$ and a finite non-pluripolar measure $\nu $ on $X$. Let $\psi _ j\in \mathrm{PSH}(X,\omega )$ be uniformly bounded and converge in capacity to a bounded $\psi \in \mathrm{PSH}(X,\omega )$. Suppose
\[ \mu _{u_ j}\leq A\omega _{\psi _ j}^ n+\nu . \]
Then $u\in \mathcal E_\varphi $, $u_ j\to u$ in capacity, $\mu _ u$ is finite, and
\[ \mu _{u_ j}\rightharpoonup \mu _ u. \]
Proof.
Translate so that $u_ j,u\leq 0$. The functions $\mathrm{e}^{u_ j},\mathrm{e}^ u$ are uniformly bounded quasi-plurisubharmonic functions with a common lower curvature bound. They converge in $L^1$ for smooth volume, since the exponential is Lipschitz on $(-\infty ,0]$. Hence lemma 5.44.7, lemma 5.44.2 give
\[ \int _ X|\mathrm{e}^{u_ j}-\mathrm{e}^ u|\, \omega _{\psi _ j}^ n\longrightarrow 0, \qquad \int _ X|\mathrm{e}^{u_ j}-\mathrm{e}^ u|\, \mathrm{d}\nu \longrightarrow 0. \]
Thus the integration error in proposition 15.37.17 tends to zero, and that proposition gives convergence in capacity and $u\in \mathcal E_\varphi $.
By lemma 5.29.16, the bounded reference measures $\omega _{\psi _ j}^ n$ have uniformly bounded mass and satisfy a common capacity estimate. Hence, for a constant $B$ independent of $j$ and $T$,
\[ \mu _{u_ j}(\{ u_ j<-T\} ) \leq B\operatorname {Cap}_\omega (\{ h<-T\} ) +\nu (\{ h<-T\} )\longrightarrow 0 \]
uniformly in $j$. The first term tends to zero because $h$ is a quasi-plurisubharmonic function not identically $-\infty $, and the second because $\nu $ is finite and does not charge the pole set of $h$. The total masses $\mu _{u_ j}(X)$ are uniformly bounded by the assumed domination. The tail criterion for smooth backgrounds, theorem 13.15.23, therefore gives finiteness of $\mu _ u$ and the asserted weak convergence.
$\square$
Pome
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