Theorem 5.41.95EA08D4F62B2

Proof. Each term $(x,z)\mapsto \Gamma _\tau (x)+\tau \operatorname{Re}z$ of the supremum satisfies the current inequality on the product, since $\operatorname{Re}z$ is pluriharmonic. The bound $\Gamma _\tau \leq 0$ gives $U_ t\leq ct$, while any fixed $\tau _0<c$ gives the minorant $\Gamma _{\tau _0}+t\tau _0$. By the local upper-envelope theorem, as in section 5.40, the upper semicontinuous regularization $V$ of $U$ on the product is quasi-plurisubharmonic, satisfies $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}V\geq 0$, and agrees with $U$ almost everywhere. It does not depend on $\operatorname {Im}z$ and satisfies $V_ t\leq ct$. The pointwise supremum $U$ is measurable, since by the left continuity in lemma 5.41.8 the parameters may be restricted to rational numbers.

By Fubini's theorem, there is a set $Z\subset X$ of zero smooth volume, containing the pole set of $\Gamma _{\tau _0}$, such that $U_ t(x)=V_ t(x)$ for almost every $t>0$ whenever $x\notin Z$. For such $x$, both sides are finite convex functions of $t$, by lemma 5.41.5, lemma 5.41.1, so by continuity the equality holds for every $t>0$.

For each $\tau <c$, proposition 5.41.3 gives

Off $Z$ it equals $\Gamma _\tau $ by lemma 5.41.8. In particular, $\chi _\tau $ is not identically $-\infty $, and lemma 5.24.3 gives $\chi _\tau =\Gamma _\tau $ everywhere. At each $x$, the convex function $t\mapsto V_ t(x)$ can be recovered from these inverse transforms by lemma 5.41.5, also when it is identically $-\infty $. We obtain

Finally, the minorant $\Gamma _{\tau _0}+t\tau _0$ shows that no slice $U_ t$ is identically $-\infty $, so every slice belongs to $\mathrm{PSH}(X,\theta )$. $\square$