Corollary 16.3.14.B6EF8456606F Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Let
\[ \eta \coloneqq \operatorname *{sup}_{C>0}{}\! {}^{*} (\varphi +C)\land \psi .{}^{\href{#sgpt-a0000019653}{1}} \]
Assume that $\eta \not\equiv -\infty $ (for example when $\varphi \preceq \psi $). Then
\[ \theta _{\eta }^ n\leq \mathds {1}_{\{ \eta =\psi \} }\, \theta _{\psi }^ n. \]
Proof.
Thanks to 16.3.3, for each $C>0$, we have
\[ \begin{aligned} \theta _{(\varphi +C)\land \psi }^ n\leq & \mathds {1}_{\{ (\varphi +C)\land \psi =\varphi +C\} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ (\varphi +C)\land \psi =\psi \} }\, \theta _{\psi }^ n\\ \leq & \mathds {1}_{\{ \varphi +C\leq \psi \} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ \eta =\psi \} } \, \theta _{\psi }^ n. \end{aligned} \]
We wish to let $C\to \infty $. The dominated convergence theorem and the semicontinuity theorem corollary 13.13.12 give, as $C\to \infty $,
\[ \mathds {1}_{\{ \varphi +C\leq \psi \} } \theta _{\varphi }^ n\rightharpoonup 0,\quad \theta _{(\varphi +C)\land \psi }^ n\rightharpoonup \theta _{\eta }^ n. \]
So we conclude that
\[ \theta _{\eta }^ n\leq \mathds {1}_{\{ \eta =\psi \} } \, \theta _{\psi }^ n. \]
This completes the proof.
$\square$
Pome
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