Corollary 16.28.130C3E3295B18F

Proof. Let $u\in \mathcal L_\delta $ satisfy $u\leq \psi $ on $K\setminus E$, where $E$ is pluripolar. Choose $b$ as in the lemma, with $b=-\infty $ on $E$ and $b\leq 0$ on $K$, and put $m=\inf _ K\psi $. For $\epsilon >0$ define

This function belongs to $\mathcal L_\delta $. On $K\setminus E$, the inequalities $u\leq \psi $, $b\leq 0$ and $m\leq \psi $ give $u_\epsilon \leq \psi $, and on $E$ the pole of $b$ gives $u_\epsilon =-\infty $. Thus $u_\epsilon $ is a competitor for $V_\delta $, so $u_\epsilon \leq F_\delta $; moreover, $u_\epsilon \to u$ almost everywhere as $\epsilon \downarrow 0$. By lemma 5.24.3, $u\leq F_\delta $ everywhere. Taking the supremum over $u$ and regularizing proves the asserted equalities.

Alternatively, apply proposition 5.40.6 on $\mathbb P^ n$ with the big background $\theta _\delta $ and the bounded obstacle $\psi -\delta \chi /2$, and then use (16.28.11). $\square$