Lemma 5.15.204014761D7A8A

Proof. The problem is local, so we may assume that $X$ is a domain in $\mathbb {C}^ n$ and that there is $\varphi \in \mathrm{PSH}(X)$ with $Z\subseteq \{ \varphi =-\infty \} $. It suffices to show that $\{ \varphi >-\infty \} $ is $\mathcal{F}$-dense.

Let $x\in X$ be a point with $\varphi (x)=-\infty $ and $U\subseteq X$ be an $\mathcal{F}$-open neighborhood of $x$ in $X$. We need to show that $U\cap \{ \varphi >-\infty \} \neq \varnothing $.

Thanks to theorem 5.15.11, after shrinking $U$, we may assume that there is $\psi \in \mathrm{PSH}(X)$ such that $U=\{ \psi >0\} $. Observe that $U$ is not a pluripolar set: Otherwise, $\psi \leq 0$ almost everywhere by proposition 5.14.4, and hence everywhere by proposition 5.14.6. So $\varphi |_ U\not\equiv -\infty $. We conclude that $U\cap \{ \varphi >-\infty \} \neq \varnothing $, as desired. $\square$