Proposition 10.31.26AABBF080DAD2

Proof. For $C\geq 0$ put $p_ C=P_\theta (\varphi +C,z)$. The preceding proposition gives $p_0\in \mathcal E_\varphi $. Since $p_0\leq p_ C\preceq \varphi $, upward stability gives $p_ C\in \mathcal E_\varphi $. The envelope identity, together with $p_ C\leq z$, yields the uniform bound

Consequently the increasing regularized limit $Q$ is not identically $-\infty $, belongs to $\mathcal E_\varphi $ and is at most $z$. The monotone convergence holds in capacity by lemma 5.29.19.

Finally, lemma 13.19.4, applied with $u=\varphi $ and $w=z$, gives