Proposition 17.16.65D2A9C4885B8

Proof. Convexity in $t$ and the upper limits at the endpoints bound every competitor by $(1-t)u_0+t u_1$. This bound is jointly upper semicontinuous on the interior and locally bounded above, so it persists after regularization. The local envelope theorem of section 5.40 shows that $U$ is quasi-plurisubharmonic with $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}U\geq 0$, and $U$ is still independent of $\Im z$. Since it dominates an admissible competitor, each slice belongs to $\mathrm{PSH}(X,\theta )$. The chord bound gives the joint upper limits at the endpoints, by the argument at the end of the proof of proposition 17.16.3. The envelope therefore belongs to its defining family and is its largest member.

If a competitor $w_ t$ runs from $u_0$ to $u_1$, then

In particular, almost everywhere,

which proves $u_ t^{\mathrm P}\to u_0$ in $L^1$ as $t\downarrow 0$. The argument at $t=1$ is the same.

For $0\leq a<b\leq 1$, let $v_ t$ be the Perron envelope on $(a,b)$ between $u_ a^{\mathrm P}$ and $u_ b^{\mathrm P}$. The original path restricted to $(a,b)$ runs between these endpoints and is admissible by proposition 17.16.3. Thus $v_ t$ also runs between them and dominates $u_ t^{\mathrm P}$. The joint upper limits of $v_ t$ at $a$ and $b$ allow us to glue its complexification to $U$ outside the smaller strip, by lemma 5.24.2; if $a=0$ or $b=1$, the same upper limits show that the glued family is still admissible for $u_0,u_1$. The resulting competitor cannot exceed $U$, which proves $v_ t=u_ t^{\mathrm P}$ on $(a,b)$. $\square$