Theorem 18.4.27F20FD198864

Proof. Write $q=P_\theta [u]_{\mathcal I}$. By Choquet's lemma, some sequence from the defining family has regularized supremum $q$. Replacing its terms by successive finite maxima, we obtain by lemma 5.37.4 an increasing sequence $v_ j$ that remains in the family and converges to $q$ almost everywhere.

Fix $x\in X$ and $\lambda >0$. On a coordinate ball about $x$, choose one smooth function $r$ such that every $v_ j+r$ and $q+r$ is plurisubharmonic; this is possible because all these functions have complex Hessian bounded below by the same smooth form $-\theta $. The increasing-limit assertion of theorem 5.37.3 gives

for all large $j$; the second equality holds because smooth additions do not change multiplier ideals. Hence $q\sim _{\mathcal I}u$.

Since $q\leq 0$, it belongs to its own defining family. The pointwise supremum is therefore at least $q$, and it is at most its regularization $q$; this proves the assertion about regularization. The defining families for $P_\theta [q]_{\mathcal I}$ and $P_\theta [u]_{\mathcal I}$ coincide, which proves idempotence. Finally, if $\sup _ Xq<0$, then $q-\sup _ Xq$ would be a larger competitor, which is impossible. $\square$