Theorem 18.32.204BABC8924B6

Proof. By theorem 18.4.2, $q\leq 0$ and $q\sim _{\mathcal I}\varphi $. Hence $q+\min _ Kw$ belongs to the defining family. Lemma 5.40.1 bounds the entire family above by $C$, so its regularized supremum $E$ is $\theta $-plurisubharmonic, and $E\geq q+\min _ Kw$.

For the upper bound, let $a$ be a member of the family. Since $\sup _ Xa\leq C$, proposition 18.4.4, proposition 18.4.6 give

Taking the regularized supremum over $a$ proves $E\leq q+C$.

The two bounds show that $E-q$ is bounded. Hence $E$ and $q$ have the same multiplier ideals, and these are the multiplier ideals of $\varphi $.

It remains to show that $E$ itself satisfies the obstacle condition, which is not automatic after regularization. By Choquet's lemma, $E$ is the regularization of the supremum of a countable subfamily. Replace its members by their successive finite maxima. Finite maxima are again members of the family: the multiplier-ideal order is preserved by theorem 5.37.3, and a finite union of pluripolar exceptional sets is pluripolar. We obtain an increasing sequence $a_ j$ in the family whose limit has regularization $E$. Outside a countable union of pluripolar subsets of $K$, every $a_ j$ is at most $w$, and hence so is $\lim _ ja_ j$. By the local upper-envelope theorem recalled in section 5.40, $E$ differs from $\lim _ ja_ j$ only on a pluripolar set. Thus $E\leq w$ quasi-everywhere on $K$.

We have shown that $E$ belongs to its defining family. It dominates every member by construction, so it is the largest member. In particular, the supremum in (18.32.1) equals $E$ at every point even before regularization. $\square$