Lemma 18.28.7`0F9B18404274`

Lemma 18.28.7.0F9B18404274 Suppose that $f\colon K\rightarrow (-\infty ,\infty ]$ is bounded from below, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

96B9EFFFBDA7 18.28.8

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

Proof. By our assumption, we can find $C>0$ so that $\varphi |_ K-C\leq f$. Then if $\eta $ is a candidate in (18.28.3), $\eta \lor (\varphi -C)$ is also a candidate, and $\eta \lor (\varphi -C)\in \mathrm{PSH}(X,\theta )_{>0}$. Therefore, $\eta \lor (\varphi -C)$ is also a candidate for the right-hand side of (18.28.8). The equality (18.28.8) follows. $\square$

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@misc{pome-tag-0F9B18404274,
  author = {{Pome contributors}},
  title = {Lemma 18.28.7},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/0F9B18404274/}
}

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Lemma 18.28.70F9B18404274

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Lemma 18.28.7`0F9B18404274`