Theorem 16.28.569A5902CF8C5

Proof. The local envelope theorem, applied to a countable family realizing the supremum (Choquet's lemma), gives $p\leq f$ outside a pluripolar set. Choose $M>\sup _ Xp$ and replace $f$ by $\min (f,M)$. This does not change the envelope, since $p$ itself is a competitor for the smaller obstacle. Since $p<M$, the contact set is also unchanged, so it suffices to prove the theorem for the smaller obstacle; that is, we may assume that $f$ is bounded above.

Set $f_ j=\max (f,-j)$ and $p_ j=P_\theta (f_ j)$. The constants $-j$ are competitors, so $p_ j$ is bounded. Theorem 16.28.4 gives $p_ j\leq f_ j$ quasi-everywhere and

The sequence $p_ j$ decreases and stays above $p$. Its limit is therefore $\theta $-plurisubharmonic, it lies below $f$ quasi-everywhere, and so it is a competitor for $p$. Hence $p_ j\downarrow p$.

Fix an integer $k\geq 1$ and $C>0$, and define the bounded nonnegative quasi-continuous function

Its values on pluripolar sets are immaterial. Where this coefficient is positive, $p_ k<f$ and $p>-C$. For $j\geq k$, we then have $p_ j\leq p_ k<f\leq f_ j$ and $p_ j\geq p>-C$ there. Plurifine locality and theorem 16.28.4 therefore imply

The truncated potentials are uniformly bounded and decrease to $\max (p,-C)$, so their measures converge weakly by bounded Bedford–Taylor continuity. The uniform capacity bounds and the weighted convergence in section 5.29 also allow testing against $H_{k,C}$. Hence the last identity holds with $p$ in place of $p_ j$. By plurifine locality, this says that the extended measure of $p$ vanishes on $\{ p_ k<f\} \cap \{ p>-C\} $. Off the pole set of $p$, these sets increase to $\{ p<f\} $ as first $k$ and then $C$ tend to infinity. Continuity of positive measures along increasing sequences proves the assertion, even if the total extended mass is infinite. $\square$