PomeTheorem 16.3.12.553335F185CE Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then
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16.3.13
\begin{equation} \theta _{P_{\theta }[\varphi ]}^ n\leq \mathds {1}_{\{ P_{\theta }[\varphi ]=0\} }\, \theta ^ n. \end{equation}
Theorem 16.3.12553335F185CE
Proof. Thanks to 16.3.3, for each $C>0$, we have
\[ \begin{aligned} \theta _{(\varphi +C)\land V_{\theta }}^ n\leq & \mathds {1}_{\{ (\varphi +C)\land V_{\theta }=\varphi +C\} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ (\varphi +C)\land V_{\theta }=V_{\theta }\} }\, \theta _{V_{\theta }}^ n\\ \leq & \mathds {1}_{\{ \varphi +C\leq V_{\theta }\} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ P_{\theta }[\varphi ]=V_{\theta }\} } \, \theta _{V_{\theta }}^ n. \end{aligned} \]
We wish to let $C\to \infty $. The dominated convergence theorem and the semicontinuity theorem corollary 13.13.12 give, as $C\to \infty $,
\[ \mathds {1}_{\{ \varphi +C\leq V_{\theta }\} } \theta _{\varphi }^ n\rightharpoonup 0,\quad \theta _{(\varphi +C)\land V_{\theta }}^ n\rightharpoonup \theta _{P_{\theta }[\varphi ]}^ n. \]
So we conclude that
\[ \theta _{P_{\theta }[\varphi ]}^ n\leq \mathds {1}_{\{ P_{\theta }[\varphi ]=V_{\theta }\} } \, \theta _{V_{\theta }}^ n. \]
Taking corollary 13.13.61 into consideration, we conclude (16.3.13). $\square$
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