Lemma 16.8.11.4A5CF38738F2 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class. Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then
295DE82294BA
16.8.12
\begin{equation} P_{\theta }[\varphi \lor \psi ]=P_{\theta }\Bigl [ P_{\theta }[\varphi ]\lor P_{\theta }[\psi ] \Bigr ]. \end{equation}
Proof.
Since $\varphi \lor \psi \preceq P_{\theta }[\varphi ]\lor P_{\theta }[\psi ]$, the $\leq $ direction of (16.8.12) follows. Conversely, it suffices to show that
\[ P_{\theta }[\varphi \lor \psi ]\geq P_{\theta }[\varphi ]\lor P_{\theta }[\psi ], \]
which follows from $\varphi \preceq \varphi \lor \psi $, $\psi \preceq \varphi \lor \psi $, and the monotonicity of the $P$-envelope with respect to $\preceq $.
$\square$
Pome
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