PomeLemma 17.5.10.DD59ECAA0656 Let $\theta $ be a smooth closed real $(1,1)$-form on $X$. If $q_ j,q\in \mathrm{PSH}(X,\theta )$ and $q_ j\to q$ in $L^1(X)$, then $\varlimsup _ jq_ j=q$ quasi-everywhere.
Lemma 17.5.10DD59ECAA0656
Proof. By lemma 17.4.3, the tail envelopes $a_ j=(\sup _{k\geq j}q_ k)^*$ are $\theta $-psh and decrease pointwise to $q$. For each $j$, corollary 5.47.11, applied after adding smooth local potentials of $\theta $, gives $a_ j=\sup _{k\geq j}q_ k$ outside a pluripolar set. The countable union of these sets is pluripolar (theorem 5.47.8), and outside it $q=\lim _ ja_ j=\varlimsup _ jq_ j$. $\square$
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