Lemma 15.12.45A797D7301BB

Proof. Let $v$ be the bounded solution of $\mathrm{MA}_\theta (v)=\mathrm{e}^{\lambda v}f\, \mathrm{d}V$ given by theorem 15.3.3, and write $M=\sup _ Xv$ and $a=v-M\leq 0$. The rooftop envelope $p=P_\theta (a,\varphi )$ satisfies $\varphi -\| a\| _\infty \leq p\leq \varphi $, so $p\sim \varphi $. By the rooftop inequality of proposition 16.28.6,

The second term vanishes: by proposition 16.28.7, the measure of $\varphi $ is carried by $\{ \varphi =0\} $, whereas $\varphi <a\leq 0$ on the set in question. On the first contact set, $a=p$ and $\mathrm{MA}_\theta (a)=\mathrm{e}^{\lambda (a+M)}f\, \mathrm{d}V$. Thus $\mathrm{MA}^{\mathrm{ext}}_\theta (p)\leq \mathrm{e}^{\lambda (p+M)}f\, \mathrm{d}V$, and $w=p+M$ has the required properties. $\square$