Proposition 16.28.6.107D3E29D21B Let $\theta $ be a smooth real $(1,1)$-form that is semipositive or big, and let $u,v\in \mathrm{PSH}(X,\theta )$. If $p=P_\theta (u,v)$ is not identically $-\infty $, then
\[ \mathrm{MA}_\theta (p) \leq \mathbf1_{\{ p=u<v\} }\mathrm{MA}_\theta (u) +\mathbf1_{\{ p=v\} }\mathrm{MA}_\theta (v). \]
Proof.
The envelope lies below both obstacles quasi-everywhere, hence everywhere, since the obstacles are upper semicontinuous. By the contact theorem (theorem 16.28.5 or theorem 15.8.6), its measure is carried by $\{ p=\min (u,v)\} $. Partition this set into $\{ p=u<v\} $ and $\{ p=v\} $, and apply proposition 13.14.6 on each part. All measures vanish on the pluripolar pole sets, which can therefore be discarded throughout.
$\square$
Pome
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