Theorem 5.13.19 (Josefson's theorem)DB05CEAF06C1

Proof. Apply Josefson's theorem, proved earlier in the collection to $E$. It constructs a function $V\in\mathrm{PSH}(\mathbb C^n)$ with $E\subset\{V=-\infty\}$, in fact with the additional bound $V(z)\leq\log^+|z|$. Taking $\varphi=V$ gives the assertion. The earlier construction is a proof of this stronger statement, so no additional external theorem is being invoked here. In dimension zero a pluripolar set is empty and a finite constant suffices.