Lemma 16.28.6918D2E34A9D42

Proof. The obstacle $u-q$ is nonnegative away from the poles, so by lemma 5.40.2 the envelope $p=P_{(1-a)\theta }(u-q)$ is bounded. By the contact theorem, its measure is carried by $D=\{ p+q=u\} $, and $p+q\leq u$ everywhere. Since $p+q$ is $\theta $-plurisubharmonic, positivity of mixed products and ordered contact imply

where all assertions are first restricted to the complement of the poles. Writing $p=(1-a)\widetilde p$ with $\widetilde p\in \mathrm{PSH}(X,\theta )\cap L^\infty (X)$ and integrating, we obtain the lower bound from the definition of $v_-(\theta )$. Finally, the bounded potential $p$ has strictly positive mass by proposition 16.28.55, since $(1-a)\theta $ is big. $\square$