Corollary 16.8.15564E8A0E68F9

Proof. Note that $\psi \sim _ P \varphi $ implies that $\psi \in \mathrm{PSH}(X,\theta )_{>0}$ by proposition 16.3.10. We observe that

since for any candidate $\psi $ for the first supremum, $\psi \lor (\varphi -\sup _ X \varphi )$ is larger than $\psi $, and is a candidate for the second supremum thanks to 16.8.13. So the first equality is a direct consequence of proposition 16.8.7 and theorem 16.3.15.

Next we prove the second equality. We only need to show that for any $\psi \in \mathrm{PSH}(X,\theta )$ with $\psi \leq 0$ and $\psi \preceq _ P \varphi $, we have $\psi \leq P_{\theta }[\varphi ]$.

By 16.8.13 and example 16.8.6, we know that $P_{\theta }[\varphi ]\lor \psi \sim _ P \varphi $ and $P_{\theta }[\varphi ]\lor \psi \leq 0$. It follows from the first equality that $\psi \leq P_{\theta }[\varphi ]$. $\square$