Lemma 16.3.3.4CF7246E6E5A Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi \land \psi \in \mathrm{PSH}(X,\theta )$. Then
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16.3.4
\begin{equation} \theta _{\varphi \land \psi }^ n\leq \mathds {1}_{\{ \varphi \land \psi =\varphi \} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ \varphi \land \psi =\psi \} } \, \theta _{\psi }^ n. \end{equation}
Proof.
We first observe that as a consequence of 13.13.14, we have
\[ \mathds {1}_{\{ \varphi \land \psi =\varphi \} } \, \theta _{\varphi \land \psi }^ n\leq \mathds {1}_{\{ \varphi \land \psi =\varphi \} } \, \theta _{\varphi }^ n,\quad \mathds {1}_{\{ \varphi \land \psi =\psi \} } \, \theta _{\varphi \land \psi }^ n\leq \mathds {1}_{\{ \varphi \land \psi =\psi \} } \, \theta _{\psi }^ n. \]
Applying theorem 13.13.24 to $\min \{ \varphi ,\psi \} $, we conclude that
\[ \theta _{\varphi \land \psi }^ n\leq \mathds {1}_{\{ \varphi \land \psi =\varphi \} } \, \theta _{\varphi \land \psi }^ n+\mathds {1}_{\{ \varphi \land \psi =\psi \} }\, \theta _{\varphi \land \psi }^ n\leq \mathds {1}_{\{ \varphi \land \psi =\varphi \} } \, \theta _{\varphi }^ n+\mathds {1}_{\{ \varphi \land \psi =\psi \} } \, \theta _{\psi }^ n, \]
and (16.3.4) is established.
$\square$
Pome
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