Corollary 15.9.4.F856D22ADF4C In the setting of theorem 15.9.1, with the bounds $\varphi -C\leq u_ j\leq \varphi +C$, suppose either
$\mathrm{MA}^{\mathrm{ext}}_\theta (u_ j)\leq \nu $ for a fixed finite non-pluripolar measure $\nu $; or
$\mathrm{MA}^{\mathrm{ext}}_\theta (u_ j)\leq A\omega _{w_ j}^ n$, where $A$ is fixed and $w_ j\in \mathrm{PSH}(X,\omega )$ are uniformly bounded and converge in capacity to a bounded $w$.
Then $u_ j\to u$ in capacity, all the measures $\mathrm{MA}^{\mathrm{ext}}_\theta (u_ j)$ and $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ have finite mass, and
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (u_ j) \rightharpoonup \mathrm{MA}^{\mathrm{ext}}_\theta (u). \]
Proof.
In the first case, lemma 5.44.2, applied to the exponentials $\mathrm{e}^{u_ j}$, proves (15.9.2). In the second case we use lemma 5.44.7 instead, since these exponentials are bounded and have a common lower bound for their curvature. In both cases the criterion gives convergence in capacity.
The masses are uniformly bounded by those of the dominating measures. For the deep sublevels we use $u_ j\geq \varphi -C$, which gives $\{ u_ j<-T\} \subset \{ \varphi <C-T\} $. In the first case the masses of these sublevels are at most $\nu (\{ \varphi <C-T\} )$, which tends to zero as $T\to \infty $ because $\nu $ does not charge the pole set of $\varphi $. In the second case lemma 5.29.16 bounds them by $C'\operatorname {Cap}_\omega (\{ \varphi <C-T\} )$, which also tends to zero. The tail criterion theorem 13.15.13, in the general-background form of theorem 13.15.23, now gives finiteness of the limiting measure and weak convergence.
$\square$
Pome
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