Lemma 13.13.14.F56651914DE1 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then
37A13FA36078
13.13.15
\begin{equation} \theta _{\varphi \lor \psi }^ n\geq \mathds {1}_{\{ \varphi \geq \psi \} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ \varphi <\psi \} }\, \theta _{\psi }^ n. \end{equation}
In particular, if $\varphi \leq \psi $, then
\[ \mathds {1}_{\{ \varphi =\psi \} }\, \theta _{\varphi }^ n\leq \mathds {1}_{\{ \varphi =\psi \} }\, \theta _{\psi }^ n. \]
Proof.
For each $k\in \mathbb {N}$, we set
\[ \psi _ k=\psi \lor (V_{\theta }-k),\quad \varphi _ k=\varphi \lor (V_{\theta }-k). \]
For each $t>0$ and each $k\in \mathbb {N}$, we have
\[ \begin{split} \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\geq & \mathds {1}_{\{ \psi _ k>\varphi _ k+t\} }\, \theta _{\psi _ k\lor (\varphi _ k+t)}^ n+\mathds {1}_{\{ \psi _ k<\varphi _ k+t\} }\, \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\\ =& \mathds {1}_{\{ \psi _ k>\varphi _ k+t\} }\, \theta _{\psi _ k}^ n+\mathds {1}_{\{ \psi _ k<\varphi _ k+t\} }\, \theta _{\varphi _ k}^ n, \end{split} \]
where the equality follows from the plurifine locality of non-pluripolar products proposition 13.11.4 1. We observe that for any $t>0$, $\psi _ k\lor (\varphi _ k+t)\sim \psi _ k\lor \varphi _ k$. Therefore, the monotonicity theorem theorem 13.13.11 guarantees that
\[ \int _ X \theta _{\psi _ k\lor (\varphi _ k+t)}^ n=\int _ X\theta _{\psi _ k\lor \varphi _ k}^ n. \]
As $t\to 0+$,
\[ \theta _{\psi _ k\lor (\varphi _ k+t)}^ n\rightharpoonup \theta _{\psi _ k\lor \varphi _ k}^ n \]
by theorem 13.13.9.
Letting $t\to 0+$, we conclude that
\[ \theta _{\psi _ k\lor \varphi _ k}^ n\geq \mathds {1}_{\{ \psi _ k>\varphi _ k\} }\, \theta _{\psi _ k}^ n+\mathds {1}_{\{ \psi _ k\leq \varphi _ k\} }\, \theta _{\varphi _ k}^ n. \]
In particular, multiplying both sides by $\mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} }$ and applying proposition 13.11.4 1 again, we find that
\[ \begin{split} \mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} }\, \theta _{\psi \lor \varphi }^ n\geq & \mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} \cap \{ \psi \leq \varphi \} }\, \theta _{\varphi }^ n \\ & +\mathds {1}_{\{ \min \{ \varphi ,\psi \} >V_{\theta }-k\} \cap \{ \psi >\varphi \} }\, \theta _{\psi }^ n. \end{split} \]
Letting $k\to \infty $, we conclude (13.13.15).
$\square$
Pome
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