Proof.
Replacing simultaneously $\gamma ,\varphi _ j,\varphi $ by $\gamma -B,\varphi _ j-B,\varphi -B$ for $B\gg 1$, we may assume that $\gamma \leq 0$. The assumptions and the conclusion are invariant under this translation. Since $\gamma \in \mathcal E(X,\theta ;\phi )$ and $P_\theta [\phi ]=\phi $, we then have $\gamma \leq \phi $.
Step 1. Assume that $(\varphi _ j)_ j$ is decreasing. In this case, we prove
B09C34234351
16.5.47
\begin{equation} \int _ X (\varphi -\gamma )\, \theta _{\varphi }^ n\geq -4A. \end{equation}
By 16.5.26, for any $j, k>0$,
\[ \int _ X (\varphi _ j-\gamma )\, \theta _{\varphi _ k}^ n\geq -4A. \]
For any $C>0$,
\[ \int _ X \Bigl (\varphi _ j \lor (\gamma -C)-\gamma \Bigr )\, \theta _{\varphi _ k}^ n\geq \int _ X (\varphi _ j-\gamma ) \, \theta _{\varphi _ k}^ n\geq -4A. \]
Letting $k\to \infty $, by theorem 13.13.9, we find
\[ \int _ X \Bigl (\varphi _ j \lor (\gamma -C)-\gamma \Bigr )\, \theta _{\varphi }^ n\geq -4A. \]
Letting $j\to \infty $, by the monotone convergence theorem, we get
\[ \int _ X \Bigl (\varphi \lor (\gamma -C)-\gamma \Bigr )\, \theta _{\varphi }^ n\geq -4A. \]
Then we let $C\to \infty $, again by the monotone convergence theorem, we conclude (16.5.47).
Step 2. In general, let
\[ \psi _ j=\operatorname *{sup}_{k\geq j}{}\! {}^{*} \varphi _ k. \]
Then $\varphi =\inf _{j>0}\psi _ j$ due to corollary 5.14.8.
For each $C>0$, let
\[ \psi _{j,C}=\psi _ j \lor (\gamma -C),\quad \varphi _ C=\varphi \lor (\gamma -C). \]
Then $\psi _{j,C},\varphi _ C\in \mathcal{E}(X,\theta ;\phi )$ since $\gamma -C\in \mathcal{E}(X,\theta ;\phi )$ thanks to the monotonicity theorem theorem 13.13.11. Observe that $\psi _{j,C}$ decreases to $\varphi _ C$ as $j\to \infty $. Moreover,
\[ \gamma \geq \psi _{j,C} \geq \psi _ j\geq \varphi _ j. \]
By 16.5.43,
\[ \int _ X (\psi _{j,C}-\gamma )\, \theta _{\psi _{j,C}}^ n\geq -2^{n+1}A. \]
By Step 1,
053AD38249AF
16.5.48
\begin{equation} \int _ X(\varphi _ C-\gamma )\, \theta _{\varphi _ C}^ n\geq -2^{n+3}A. \end{equation}
In particular, by the plurifine locality of the non-pluripolar product proposition 13.11.4 1,
\[ \int _{\{ \varphi >\gamma -C\} }(\varphi -\gamma )\, \theta _{\varphi }^ n\geq -2^{n+3}A. \]
Letting $C\to \infty $, we conclude (16.5.46). As a consequence,
\[ \int _ X (\gamma -\varphi )\, \theta _{\varphi }^ n<\infty . \]
We still have to prove that $\varphi \in \mathcal{E}(X,\theta ;\phi )$. In fact, by (16.5.48), for any $C>0$,
\[ \int _{\{ \varphi \leq \gamma -C\} }\, \theta _{\varphi _ C}^ n\leq \frac{1}{C}\int _ X (\gamma -\varphi _ C)\, \theta _{\varphi _ C}^ n\leq 2^{n+3}C^{-1}A. \]
Using the monotonicity theorem theorem 13.13.11 and the plurifine locality of the non-pluripolar product proposition 13.11.4 1, we find
\[ \int _ X \theta _{\phi }^ n=\int _ X \theta _{\varphi _ C}^ n=\int _{\{ \varphi \leq \gamma -C\} }\, \theta _{\varphi _ C}^ n+\int _{\{ \varphi >\gamma -C\} }\, \theta _{\varphi }^ n. \]
Letting $C\to \infty $, we conclude that
\[ \int _ X \theta _{\varphi }^ n=\int _ X \theta _{\phi }^ n. \]
This completes the proof.
$\square$
Pome
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