Theorem 15.8.66F087EC35E2E

Proof. As in the proof of theorem 16.28.5, the local envelope theorem and a countable Choquet family give $p\leq f$ quasi-everywhere. Set $f_ j=\max (f,-j)$ and $p_ j=P_\theta (f_ j)$. Then $p$ is a competitor for every $f_ j$. Since $f_ j\leq f_1$ and $f_1$ is finite almost everywhere, the finite-sublevel argument in section 5.40 gives a common upper bound for all competitors. Thus the envelopes $p_ j$ are $\theta $-plurisubharmonic and uniformly bounded above. Choose $M$ strictly larger than this bound.

The obstacle $g_ j=\min (f_ j,M)$ is bounded and quasi-continuous. Since $p_ j<M$ and $p_ j\leq f_ j$ quasi-everywhere, $p_ j$ is a competitor for $P_\theta (g_ j)$; the reverse inequality follows from $g_ j\leq f_ j$. Thus $p_ j=P_\theta (g_ j)$, and proposition 15.8.5 gives contact support for $p_ j$ with respect to $g_ j$. Because $p_ j<M$, no contact occurs at the artificial upper cutoff, so

Each $p_ j$ is locally bounded off $Z$, with a lower barrier of the form $\rho -C_ j$. The sequence decreases to $p$: its limit is not identically $-\infty $ because it dominates $p$, and it lies below $f$ quasi-everywhere, so it is itself a competitor for $P_\theta (f)$.

Fix a Hermitian form $\Omega \geq \theta $. For integers $C,k\geq 1$, consider the bounded $\Omega $-plurisubharmonic functions

and set

extended by zero on $Z$. This function is bounded and quasi-continuous. Indeed, off $Z$ the function $p_ k$ is a finite quasi-plurisubharmonic function, truncation by bounded continuous functions preserves quasi-continuity, and the pluripolar set $Z$ can be removed by an open set of arbitrarily small capacity. At a pole of $p$ the second factor vanishes.

For $j\geq k$, on $\{ H_{k,C}>0\} $ we have, outside pluripolar sets,

By plurifine locality, the signed measure $\nu _{j,C}=(\theta +\mathrm{dd}^{\mathrm{c}}v_{j,C})^ n$ therefore agrees there with $\mathrm{MA}_\theta (p_ j)$, and (15.8.7) gives

For fixed $C$, the potentials $v_{j,C}$ have common global bounds and decrease to $v_ C$, so lemma 5.29.17 yields

On the set where $H_{k,C}$ is positive we have $p>-C$, so the restricted product in theorem 13.14.3 identifies $H_{k,C}(\theta +\mathrm{dd}^{\mathrm{c}}v_ C)^ n$ with the positive measure $H_{k,C}\mathrm{MA}_\theta (p)$; on the complement the coefficient is zero. Hence $\mathrm{MA}_\theta (p)$ vanishes on $\{ H_{k,C}>0\} $.

Finally, if $p(x)$ is finite and $p(x)<f(x)$, then $p_ k(x)<f(x)$ for large $k$ and $p(x)>-C$ for large $C$. The countable union of the sets $\{ H_{k,C}>0\} $ therefore covers $\{ p<f\} $ outside $Z\cup \{ p=-\infty \} $. Both excluded sets are pluripolar, and the theorem follows. $\square$