PomeDefinition 18.28.2.28F090ED5EFA Given a function $f\colon K\rightarrow [-\infty ,\infty ]$ and $\varphi \in \mathrm{PSH}(X,\theta )$, we introduce the relative $P$-envelope of $\varphi $ (with respect to $K,f,\theta $) as
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18.28.3
\begin{equation} P_{\theta ,K}[\varphi ](f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq \varphi \Bigr \} . \end{equation}
Similarly, we define the relative $\mathcal{I}$-envelope of $\varphi $ (with respect to $K,f,\theta $) as
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18.28.4
\begin{equation} P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _{\mathcal{I}} \varphi \Bigr \} . \end{equation}
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