PomeExample 18.28.5.D0D0EA7C293D When $K=X$ and $f=0$, we have
\[ P_{\theta ,X}[\varphi ](0)=P_{\theta }[\varphi ],\quad P_{\theta ,X}[\varphi ]_{\mathcal{I}}(0)=P_{\theta }[\varphi ]_{\mathcal{I}}. \]
When $K=X$, we have
\[ P_{\theta ,X}[\varphi ](f)=P_{\theta }[\varphi ](f). \]
These definitions reduce to the envelopes studied in section 16.1. On the other hand, when $\varphi =V_{\theta }$, we have
\[ \begin{aligned} P_{\theta ,K}[V_{\theta }](f) & =P_{\theta ,K}[V_{\theta }]_{\mathcal{I}}(f)\\ & =\operatorname *{sup}_{}{}\! {}^{*}\left\{ \eta \in \mathrm{PSH}(X,\theta ): \begin{array}{l} \eta |_ K \leq f \textup{ quasi-everywhere} \end{array} \right\} . \end{aligned} \]
In an even more special case where $\varphi =V_{\theta }$ and $K=X$, we find that
\[ P_{\theta ,X}[V_{\theta }](f)=\operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta \leq f \textup{ quasi-everywhere}\Bigr \} . \]
This is nothing but $P_{\theta }(f)$ as we studied in definition 13.13.18.
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