Proposition 18.28.359724C0B32032

Proof. We only give the proof for the relative $\mathcal{I}$-envelopes; the proof for the relative $P$-envelopes is identical, replacing $P_{\theta }[\bullet ]_{\mathcal{I}}$ by $P_{\theta }[\bullet ]$ and using corollary 18.28.13 for the final candidate condition.

Set

By 18.28.15, $d_ S(\eta _ j,P_{\theta }[\varphi _ j]_{\mathcal{I}})=0$ and $d_ S(\eta ,P_{\theta }[\varphi ]_{\mathcal{I}})=0$ for all $j$. Moreover, theorem 16.9.49 gives

in case 1, and the same convergence holds in case 2 by corollary 16.9.12 and theorem 16.9.49. Consequently, $\eta _ j\xrightarrow {d_ S}\eta $ in both cases.

Assume first that $(\varphi _ j)_ j$ is decreasing. Then $(\eta _ j)_ j$ is decreasing and $\eta _ j\geq \eta $. Let $\eta _\infty \coloneqq \inf _ j \eta _ j$. The Monge–Ampère masses of $\eta _ j$ converge to the mass of $\eta $, by the preceding $d_ S$-convergence and theorem 16.9.30. Since $\eta _ j\geq \eta _\infty \geq \eta $, the monotonicity theorem theorem 13.13.11 gives

Hence $\eta _ j\xrightarrow {d_ S}\eta _\infty $ by corollary 16.9.14. The uniqueness of the $d_ S$-limit gives $d_ S(\eta _\infty ,\eta )=0$. Since $\eta _\infty \geq \eta $, proposition 16.9.6 implies $\eta _\infty \sim _ P \eta $. In particular, $\eta _\infty \preceq _{\mathcal{I}}\varphi $ by corollary 16.8.14. The condition $\eta _\infty \leq v$ on $K$ quasi-everywhere is inherited from the $\eta _ j$'s, so $\eta _\infty $ is a candidate for $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$. Thus $\eta _\infty \leq \eta $, and the reverse inequality was already observed. This proves the $\mathcal{I}$-part of 1.

Assume next that $\varphi _ j\nearrow \varphi $ almost everywhere. Then $(\eta _ j)_ j$ is increasing and $\eta _ j\leq \eta $. Put $\tilde\eta \coloneqq \operatorname *{sup}_{}{}\! {}^{*}_ j \eta _ j$. By corollary 18.28.11 and corollary 16.9.12, we have $\eta _ j\xrightarrow {d_ S}\tilde\eta $. The $d_ S$-convergence above therefore gives $d_ S(\tilde\eta ,\eta )=0$, hence $\tilde\eta \sim _ P \eta $ by proposition 16.9.6. In particular, $\tilde\eta \preceq _{\mathcal{I}}\varphi $ by corollary 16.8.14. Also $\tilde\eta \leq v$ on $K$ quasi-everywhere by proposition 5.14.5. Hence $\tilde\eta $ is a candidate for $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$, so $\tilde\eta \leq \eta $. Since $\eta _ j\leq \eta $ for all $j$, this proves $\eta _ j\nearrow \eta $ almost everywhere. $\square$