Proposition 15.37.17D220D741C684

Proof. The bounds pass to $u$ almost everywhere along a subsequence, and then everywhere by lemma 5.24.3; upward stability gives $u\in \mathcal E_\varphi $. After translating the entire family and its bounds by a common constant, we may assume that all potentials are nonpositive.

It suffices to prove convergence in capacity along a further subsequence of every subsequence. Extract so that $I_ j\leq 2^{-j}$ and $u_ j\to u$ almost everywhere for smooth volume, and form

All these envelopes lie above $h$ and below $\varphi +C$; hence none of them is identically $-\infty $, and all belong to $\mathcal E_\varphi $. Partition the contact set of $p_{j,N}$ according to the first index $\ell $ with $p_{j,N}=u_\ell $. Ordered contact on each part gives

Only integrals of nonnegative functions occur here, so the estimate holds even if a rooftop has infinite mass.

As $N\to \infty $, $p_{j,N}\downarrow p_ j$; as $j\to \infty $, the $p_ j$ increase, and their regularized limit $p$ is not identically $-\infty $. Both monotone limits hold in capacity by lemma 5.29.19. Moreover $h\leq p\leq u\leq \varphi +C$, where $p\leq u$ follows from the almost-everywhere convergence of the extracted $u_ j$. Thus $p\in \mathcal E_\varphi $.

Apply lemma 13.14.8, first as $N\to \infty $ and then as $j\to \infty $, with the bounded quasi-continuous weights in the preceding estimate. These weights converge in capacity because the exponential is Lipschitz on a common upper half-line. We obtain

The second equality follows off the poles, which the extended measures ignore. The zero-domination lemma lemma 15.37.9 now gives $p=u$; no finiteness of $\mu _ p$ is needed.

Finally, the upper envelopes $U_ j=(\sup _{\ell \geq j}u_\ell )^*$ decrease to $u$, by theorem 5.25.2 and the almost-everywhere convergence. Applying lemma 5.29.19 to $p_ j\uparrow u$ and $U_ j\downarrow u$, the inequalities $p_ j\leq u_ j\leq U_ j$ give convergence in capacity along this subsequence, and hence for the original sequence. $\square$