Corollary 16.8.30.21CC5153F56F Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that $\psi \preceq _ P \varphi $. Then for any $C>0$, $\psi \land (\varphi +C)\in \mathrm{PSH}(X,\theta )_{>0}$ and
42D232E5CD85
16.8.31
\begin{equation} P_{\theta }[\varphi ](\psi )=\operatorname *{sup}_{C>0}{}\! {}^{*} \psi \land (\varphi +C)=\psi . \end{equation}
Proof.
Since $\psi \preceq _ P\varphi $, by proposition 16.8.29, we get
\[ \psi \land (\varphi +C)\in \mathrm{PSH}(X,\theta )_{>0} \]
and $\psi \land (\varphi +C)\sim _ P\psi $ for every $C>0$.
Let
\[ \eta \coloneqq \operatorname *{sup}_{C>0}{}\! {}^{*}\psi \land (\varphi +C). \]
Then $\eta \sim _ P \psi $. By corollary 16.3.14, we have
\[ \theta _\eta ^ n\leq \mathds {1}_{\{ \eta =\psi \} }\, \theta _\psi ^ n. \]
The domination principle theorem 13.13.43 gives $\eta \geq \psi $. The reverse inequality is trivial, so $\eta =\psi $.
$\square$
Pome
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