Theorem 15.37.196FBF04A6D4DE

Proof. Start with an arbitrary subsequence. By lemma 15.37.13, a further subsequence has a common lower envelope $h\in \mathcal E_\varphi $, and $u\in \mathcal E_\varphi $. Inequality (15.37.2) and Hartogs' lemma give a common upper bound $u_ j\leq \varphi +C$.

After a common translation, assume $u_ j,u\leq 0$. Since $\mu _{u_ j}\leq \nu $, lemma 5.44.2 gives

Thus proposition 15.37.17 gives convergence in capacity along this subsequence. For the measures themselves, the common lower bound gives

since $\nu $ is finite and does not charge the pole set of $h\not\equiv -\infty $. The tail criterion for smooth backgrounds, theorem 13.15.23, therefore gives $\mu _{u_ j}\rightharpoonup \mu _ u$, and weak limits preserve domination by the fixed finite measure $\nu $. Every subsequence of the original sequence has a further subsequence with these conclusions and the same limit $u$, so both convergences hold for the original sequence. $\square$