Lemma 10.31.3468A4962EA99D

Proof. By lemma 5.24.3, the lower bound passes to the limit, so $h\leq a\preceq \varphi $, and upward stability gives $a\in \mathcal E_\varphi $. Start with any subsequence and extract a further one along which the integrals in the hypothesis are at most $2^{-j}$ and $a_ j\to a$ almost everywhere for smooth volume. The finite rooftops

are not identically $-\infty $ because $h$ is a competitor. Partitioning their contact sets according to the first obstacle touched gives

No relative-class assertion about these finite rooftops is needed. Their decreasing limits $p_ j=P_\theta (\inf _{k\geq j}a_ k)$ are at least $h$, and the almost-everywhere convergence gives $p_ j\leq a$. Hence the $p_ j$ and their increasing regularized limit $p$ lie in $\mathcal E_\varphi $.

The limits in $N$ and then in $j$ hold in capacity by lemma 5.29.19, since $h$ is a common lower bound. Apply the weighted lower semicontinuity lemma 13.14.8 twice, to the bounded weights $\min (1,|p_{j,N}-a|)\mathrm{e}^{\lambda a}$ and their limits. These weights converge in capacity: discard deep sublevel sets of $h$ and $a$, and use continuity on the remaining finite ranges. We obtain

The exponential is positive off a pluripolar set, which $\mu _ p$ does not charge. Hence $\mu _ p(\{ p<a\} )=0$, and zero domination gives $p=a$. The upper envelopes $(\sup _{k\geq j}a_ k)^*$ decrease to $a$, and squeezing proves convergence in capacity along this subsequence. Since the initial subsequence was arbitrary, the convergence holds for the original sequence. $\square$