Remark 5.17.40CEC6D75820C

Remark 5.17.4.0CEC6D75820C More generally, if $K\subseteq X$ is a closed non-pluripolar subset, then

is relatively compact with respect to the $L^1$-topology. See [Corollary 4.3, GZ05].

As a consequence, if $\sup _ K \varphi \leq C_0$ for some $C_0\in \mathbb {R}$, then $\sup _ X \varphi \leq C_1$ for some $C_1\in \mathbb {R}$ as well.

To see this, we may assume that $C_0>0$. When $\sup _ K \varphi \in [0,C_0]$, our assertion follows from the above. If $\sup _ K \varphi <0$, it suffices to observe that