Pome

Corollary 18.28.1372097BEFEC59

Corollary 18.28.13.72097BEFEC59 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

\[ \begin{aligned} P_{\theta ,K}[\varphi ](f)= & \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} . \end{aligned} \]

Proof. We first apply proposition 18.28.12,

\[ \begin{split} P_{\theta ,K}[\varphi ](f)= & P_{\theta ,K}\left[P_{\theta }[\varphi ]\right](f)\\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \end{split} \]

Next observe that by 18.28.7, the latter can also be written as

\[ \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \]

Next we apply 18.28.9; the remaining statements follow. $\square$

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