Pome7FE9391D7A51Corollary 16.13.5.7FE9391D7A51 Suppose that $\{ \theta \} $ is big, let $u,v\in \mathrm{PSH}(X,\theta )$, and suppose that $w=P_\theta (u,v)\ne -\infty $. Then
More generally, for finitely many potentials $u_1,\ldots ,u_ r$ such that $w=P_\theta (\min _ ju_ j)\ne -\infty $, the measure $\mu _ w$ is carried by $\bigcup _ j\{ w=u_ j\} $.
Proof. By lemma 16.11.2, potentials are quasi-continuous, and hence so is the obstacle $\min \{ u,v\} $. Its family of competitors is nonempty, because $w\ne -\infty $, and bounded above by $\sup _ Xu$. By theorem 16.13.4, $\mu _ w$ is carried by $\{ w=\min \{ u,v\} \} \subset \{ w=u\} \cup \{ w=v\} $. Since $w\leq u$ and $w\leq v$ everywhere, lemma 16.13.1 gives $\mathbf1_{\{ w=u\} }\mu _ w\leq \mathbf1_{\{ w=u\} }\mu _ u$ and $\mathbf1_{\{ w=v\} }\mu _ w\leq \mathbf1_{\{ w=v\} }\mu _ v$. Adding these inequalities proves the first assertion; the intersection of the two contact sets is counted twice, which is harmless for an upper bound. The same argument, applied to the obstacle $\min _ ju_ j$, proves the last assertion. $\square$
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