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Proposition 16.5.22E2221B6939BE

Proposition 16.5.22.E2221B6939BE Let $\varphi \in \mathcal{E}(X,\theta ;\phi )$. The following are equivalent:

  1. 0DF993971FDE

    $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$;

  2. 4FACCBDBBADB

    $E^{\phi }_{\theta }(\varphi )>-\infty $.

Proof. Fix $\varphi \in \mathcal{E}(X,\theta ;\phi )$. Without loss of generality, we may assume that $\varphi \leq 0$. In particular, $\varphi \leq P_{\theta }[\varphi ]\leq P_{\theta }[\phi ]=\phi $.

2 $\implies $ 1. Assume 2. Using the plurifine locality of the non-pluripolar product proposition 13.11.4 1, we have that $\mathds {1}_{\{ \varphi >\phi -C\} } \theta _{\varphi \lor (\phi -C)}^ n=\mathds {1}_{\{ \varphi >\phi -C\} } \theta _{\varphi }^ n$ converges strongly2 to $\theta _{\varphi }^ n$ as $C\to \infty $ by the dominated convergence theorem. Therefore, for a fixed $D>0$,

\[ \begin{aligned} \int _ X \Bigl ( \varphi \lor (\phi -D)-\phi \Bigr )\, \theta _{\varphi }^ n=& \lim _{C\to \infty } \int _{\{ \varphi >\phi -C\} } \Bigl ( \varphi \lor (\phi -D)-\phi \Bigr )\, \theta _{\varphi \lor (\phi -C)}^ n\\ \geq & \varlimsup _{C\to \infty } \int _{\{ \varphi >\phi -C\} } \Bigl ( \varphi \lor (\phi -C)-\phi \Bigr )\, \theta _{\varphi \lor (\phi -C)}^ n\\ \geq & \varlimsup _{C\to \infty } \int _ X \Bigl ( \varphi \lor (\phi -C)-\phi \Bigr )\, \theta _{\varphi \lor (\phi -C)}^ n\\ \geq & \varlimsup _{C\to \infty } (n+1)E_{\theta }^{\phi }\Bigl (\varphi \lor (\phi -C) \Bigr )\\ \geq & (n+1)E_{\theta }^{\phi }(\varphi ). \end{aligned} \]

Here the last but one inequality follows from (16.5.11) and the fact that $\varphi \lor (\phi -C)\leq \phi $. Letting $D\to \infty $, we conclude 1.

1 $\implies $ 2. Assume 1. Thanks to 16.5.16, we know that for each $C>0$,

\[ \begin{aligned} E_{\theta }^{\phi }\Bigl (\varphi \lor (\phi -C)\Bigr )\geq & \int _ X \Bigl (\varphi \lor (\phi -C)-\phi \Bigr )\, \theta _{\varphi \lor (\phi -C)}^ n\\ =& \int _{\left\{ \varphi >\phi -C \right\} } \Bigl (\varphi -\phi \Bigr )\, \theta _{\varphi \lor (\phi -C)}^ n+ \int _{\left\{ \varphi \leq \phi -C \right\} } (-C)\, \theta _{\varphi \lor (\phi -C)}^ n \\ =& \int _{\{ \varphi +C>\phi \} } \left(\varphi +C-\phi \right)\, \theta _{\varphi }^ n-C\int _ X \theta _{\phi }^ n\\ \geq & \int _ X \left(\varphi +C-\phi \right)\, \theta _{\varphi }^ n-C\int _ X \theta _{\phi }^ n\\ =& \int _ X (\varphi -\phi )\, \theta _{\varphi }^ n. \end{aligned} \]

Due to proposition 16.5.20, we have

\[ \lim _{C\to \infty }E_{\theta }^{\phi }\Bigl (\varphi \lor (\phi -C)\Bigr )=E_{\theta }^{\phi }(\varphi ), \]

so 2 holds. $\square$

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