PomeCorollary 13.15.14.7312F05A4396 Let $u_ j,u\in \mathrm{PSH}(X,\omega )$ be nonpositive. If $u_ j\to u$ in capacity and $\sup _ jJ_{\chi ,\omega }(u_ j)\leq C$, then
\[ J_{\chi ,\omega }(u)\leq C,\qquad \mu _{u_ j}\rightharpoonup \mu _ u. \]
Corollary 13.15.147312F05A4396
Proof. Since $\chi \geq 1$, the total masses are at most $C$, and
\[ \mu _{u_ j}(\{ u_ j<-T\} )\leq C/\chi (T). \]
Thus theorem 13.15.13 applies. For the bound on $J_{\chi ,\omega }(u)$, use (13.15.10) with the compactly supported continuous function $s\mapsto \chi (s)\eta _ T(s)$ in place of $w\eta _ T$. The corresponding weighted integrals converge for each fixed $T$, and they are at most $C$. Let $T\to \infty $ and use monotone convergence. $\square$
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