Proof.
We may assume that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, as otherwise (16.3.36) follows immediately from the monotonicity theorem theorem 13.13.11.
Step 1. We claim that it suffices to prove (16.3.36) with $P_{\theta }[\varphi ]$ and $P_{\theta }[\psi ]$ in place of $\varphi $ and $\psi $.
In fact, as $\varphi \land \psi \not\equiv -\infty $, we also have $P_{\theta }[\varphi ]\land P_{\theta }[\psi ]\not\equiv -\infty $. Moreover,
\[ \int _ X \theta _{P_{\theta }[\varphi ]}^ n=\int _ X \theta _{\varphi }^ n,\quad \int _ X \theta _{P_{\theta }[\psi ]}^ n=\int _ X \theta _{\psi }^ n \]
by proposition 16.3.10. On the other hand, as $C\to \infty $, the potentials
\[ \Bigl ( (\varphi +C)\land V_{\theta } \Bigr ) \lor \Bigl ( (\psi +C)\land V_{\theta } \Bigr ) \]
converge to $P_{\theta }[\varphi ]\lor P_{\theta }[\psi ]$ almost everywhere. Therefore, thanks to corollary 13.13.12 and theorem 13.13.11,
\[ \int _ X \theta _{P_{\theta }[\varphi ]\lor P_{\theta }[\psi ]}^ n=\int _ X \theta _{\varphi \lor \psi }^ n. \]
Finally, we have
\[ \int _ X \theta _{P_{\theta }[\varphi ]\land P_{\theta }[\psi ]}^ n= \int _ X \theta _{\varphi \land \psi }^ n \]
as well. When the left-hand side vanishes, this follows from theorem 13.13.11. Otherwise, it follows from 16.3.32.
In particular, (16.3.36) is equivalent to the corresponding result with $P_{\theta }[\varphi ]$ and $P_{\theta }[\psi ]$ in place of $\varphi $ and $\psi $.
Step 2. We shall assume that $\varphi $ and $\psi $ are model potentials in the sequel.
We write $\gamma =\varphi \lor \psi $ for simplicity. Note that $\gamma \leq 0$.
For each $C>0$, we introduce
\[ \varphi _ C=\varphi \lor (\gamma -C),\quad \psi _ C=\psi \lor (\gamma -C). \]
Note that $\varphi _ C\sim \psi _ C\sim \gamma $. Observe that
\[ \inf _{C>0} \varphi _ C\land \psi _ C=\varphi \land \psi . \]
The $\geq $ direction is trivial, and the $\leq $ direction follows from the observation that the left-hand side is dominated by both $\varphi _ C$ and $\psi _ C$ for any $C>0$, and therefore by both $\varphi $ and $\psi $.
For $C>0$, we compute
35F5A4FBAC68
16.3.37
\begin{equation} \begin{aligned} \theta _{\varphi _ C}^ n=& \mathds {1}_{\{ \varphi >\gamma -C\} } \, \theta _{\varphi _ C}^ n+ \mathds {1}_{\{ \varphi \leq \gamma -C\} } \, \theta _{\varphi _ C}^ n & \\ =& \mathds {1}_{\{ \varphi >\gamma -C\} } \, \theta _{\varphi }^ n+ \mathds {1}_{\{ \varphi \leq \gamma -C\} } \, \theta _{\varphi _ C}^ n & \textup{by proposition \href{../../tag/9E81D36B1290/}{13.11.4} \href{../../tag/9E81D36B1290/#sgpt-1C036AD6}{1}}\\ =& \theta _{\varphi }^ n+ \mathds {1}_{\{ \varphi \leq \gamma -C\} } \, \theta _{\varphi _ C}^ n & \textup{by theorem \href{../../tag/553335F185CE/}{16.3.12}}. \end{aligned} \end{equation}
In the last line we use that $\varphi $ is model, so $\theta _\varphi ^ n$ is carried by $\{ \varphi =0\} \subseteq \{ \varphi >\gamma -C\} $. We also get a similar formula for $\theta _{\psi _ C}^ n$.
Therefore, taking the monotonicity theorem theorem 13.13.11 into consideration, we find
BCB4335E3D6E
16.3.38
\begin{equation} \int _ X \theta _{\gamma }^ n-\int _ X \theta _{\varphi }^ n=\int _{\{ \varphi \leq \gamma -C\} } \theta _{\varphi _ C}^ n,\quad \int _ X \theta _{\gamma }^ n-\int _ X \theta _{\psi }^ n=\int _{\{ \psi \leq \gamma -C\} } \theta _{\psi _ C}^ n. \end{equation}
On the other hand, using 16.3.3 and (16.3.37), we find
\[ \begin{aligned} \theta _{\varphi _ C\land \psi _ C}^ n \leq & \mathds {1}_{\{ \varphi _ C\land \psi _ C=\varphi _ C\} } \, \theta _{\varphi _ C}^ n+\mathds {1}_{\{ \varphi _ C\land \psi _ C=\psi _ C\} } \, \theta _{\psi _ C}^ n \\ \leq & \mathds {1}_{\{ \varphi _ C\land \psi _ C=\varphi _ C=0\} }\, \theta _{\varphi }^ n+\mathds {1}_{\{ \varphi _ C\land \psi _ C=\psi _ C=0\} }\, \theta _{\psi }^ n+\mathds {1}_{\{ \varphi \leq \gamma -C\} }\, \theta _{\varphi _ C}^ n\\ & +\mathds {1}_{\{ \psi \leq \gamma -C\} }\, \theta _{\psi _ C}^ n. \end{aligned} \]
In particular,
\[ \mathds {1}_{\{ \varphi _ C\land \psi _ C<0 \} }\, \theta _{\varphi _ C\land \psi _ C}^ n \leq \mathds {1}_{\{ \varphi \leq \gamma -C\} }\, \theta _{\varphi _ C}^ n+\mathds {1}_{\{ \psi \leq \gamma -C\} }\, \theta _{\psi _ C}^ n. \]
Integrating, we find
\[ \begin{aligned} \int _{\{ \varphi _ C\land \psi _ C=0 \} } \theta _{\varphi _ C\land \psi _ C}^ n \geq & \int _ X \theta _{\gamma }^ n-\int _{\{ \varphi \leq \gamma -C\} } \theta _{\varphi _ C}^ n-\int _{\{ \psi \leq \gamma -C\} } \theta _{\psi _ C}^ n\\ =& \int _ X \theta _{\varphi }^ n+\int _ X \theta _{\psi }^ n-\int _ X \theta _{\gamma }^ n, \end{aligned} \]
where on the first line we used the fact that $\varphi _ C\land \psi _ C\sim \gamma $, while on the second line, we used (16.3.38).
Letting $C\to \infty $, and using corollary 13.13.16, we conclude (16.3.36).
$\square$
Pome
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