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Proposition 16.3.25F7C0BACE770D

Proposition 16.3.25.F7C0BACE770D Assume that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ and

E6B68D406648 16.3.26
\begin{equation} \int _ X \theta _{\varphi }^ n+\int _ X \theta _{\psi }^ n>\int _ X \theta _{\varphi \lor \psi }^ n. \end{equation}

Then $\varphi \land \psi \in \mathrm{PSH}(X,\theta )$.

Proof. Without loss of generality, we may assume that $\varphi ,\psi \leq 0$. For simplicity, we write

\[ \eta =P_{\theta }[\varphi \lor \psi ]. \]

Note that due to our assumption (16.3.26), $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, and hence so is $\varphi \lor \psi $ thanks to the monotonicity theorem theorem 13.13.11. By corollary 16.3.20, $\eta $ is a model potential, and $\int _ X\theta _\eta ^ n=\int _ X\theta _{\varphi \lor \psi }^ n>0$.

Take $C>0$ large enough, so that

E47535691487 16.3.27
\begin{equation} \int _{\{ \varphi >\eta -C\} } \theta _{\varphi }^ n+\int _{\{ \psi >\eta -C\} } \theta _{\psi }^ n>\int _ X \theta _{\eta }^ n. \end{equation}

This is possible thanks to proposition 13.11.4 4. Fix $C'>C$. Write

\[ \gamma _{C'}\coloneqq \Bigl (\varphi \lor (\eta -C')\Bigr )\land \Bigl (\psi \lor (\eta -C')\Bigr ). \]

Then observe that $\gamma _{C'}\sim \eta $, and

\[ \inf _{C'>C}\gamma _{C'}=\varphi \land \psi . \]

Suppose, for contradiction, that $\varphi \land \psi \equiv -\infty $. Then

\[ \adjustlimits \lim _{C'\to \infty } \sup _ X\gamma _{C'}=-\infty . \]

Thanks to proposition 16.3.21, for each $C'>C$,

\[ \sup _{X} \gamma _{C'}=\sup _{\{ \eta \neq -\infty \} } (\gamma _{C'}-\eta ), \]

since $\eta $ is a model potential. It follows that

74C044AB05CB 16.3.28
\begin{equation} \adjustlimits \lim _{C'\to \infty } \sup _{\{ \eta \neq -\infty \} } (\gamma _{C'}-\eta )=-\infty . \end{equation}

In particular, we can take $C'$ large enough so that

\[ \gamma _{C'}\leq \eta -C. \]

For such $C'$, we have

\[ \begin{aligned} \int _ X \theta _{\gamma _{C'}}^ n=& \int _{\{ \gamma _{C'}\leq \eta -C\} } \theta _{\gamma _{C'}}^ n \\ \leq & \int _{\left\{ \varphi \lor (\eta -C')\leq \eta -C\right\} } \theta _{\varphi \lor (\eta -C')}^ n +\int _{\left\{ \psi \lor (\eta -C')\leq \eta -C \right\} } \theta _{\psi \lor (\eta -C')}^ n\\ =& 2\int _ X \theta _{\eta }^ n-\int _{\{ \varphi >\eta -C\} }\theta _{\varphi }^ n-\int _{\{ \psi >\eta -C\} }\theta _{\psi }^ n\\ <& \int _ X \theta _{\eta }^ n, \end{aligned} \]

where the second line follows from 16.3.3, the fourth line follows from (16.3.27). This contradicts the fact that $\gamma _{C'}\sim \eta $ in view of theorem 13.13.11. $\square$

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