PomeCorollary 5.15.21.31032260EF62 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Set
\[ W=\left\{ x\in X:\varphi (x)=-\infty \right\} \textup{ or }W=\left\{ x\in X:\psi (x)=-\infty \right\} . \]
Then for any pluripolar set $Z\subseteq X$, we have
\[ \sup _{X\setminus W}(\varphi -\psi ) =\sup _{X\setminus (W\cup Z)}(\varphi -\psi ),\quad \inf _{X\setminus W}(\varphi -\psi ) =\inf _{X\setminus (W\cup Z)}(\varphi -\psi ). \]
In particular, taking $\psi =0$, we find that
\[ \sup _{X\setminus Z}\varphi =\sup _ X \varphi . \]
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