Proposition 15.8.5BDE06B833BC8

Proof. By lemma 5.29.15, there are bounded lower semicontinuous $f_ j$ with common bounds and $f_ j\downarrow f$ quasi-everywhere. The envelopes $p_ j=P_\theta (f_ j)$ decrease to $p=P_\theta (f)$. Indeed, the common lower barrier shows that the limit is not identically $-\infty $; the limit lies below $f$ quasi-everywhere and is thus a competitor for $P_\theta (f)$, while monotonicity gives the other inequality. All these functions have common local bounds off $Z$.

For fixed $k$, put $H_ k=\min \{ 1,(f-p_ k)_+\} $ on $X\setminus Z$, a bounded quasi-continuous function there. If $j\geq k$, then on $\{ H_ k>0\} $ we have $p_ j\leq p_ k<f\leq f_ j$ outside a pluripolar set, so proposition 15.8.4 gives $H_ k\mathrm{MA}_\theta (p_ j)=0$ off $Z$.

Let $g$ be continuous with compact support in $X\setminus Z$. The continuity of bounded products against quasi-continuous weights from section 5.27 applies to the fixed quasi-continuous coefficient $gH_ k$. Consequently $H_ k\mathrm{MA}_\theta (p)=0$ off $Z$. As $k\to \infty $, the sets $\{ H_ k>0\} $ exhaust $\{ p<f\} \setminus Z$, and $Z$ is not charged because it is pluripolar. $\square$