Lemma 13.23.12B568CD72F9D

Proof. Whether a finite rooftop envelope is identically $-\infty $ does not change under bounded changes of its obstacles. By the definition of $U(u_ i)$, we can choose $v_ i\in \mathrm{PSH}(X,\theta )$ with $v_ i\sim u_ i$ whose masses satisfy

Translate so that $v_ i\leq w\leq 0$, and put

The common competitor $w-k$ gives $w-k\leq p_ k\leq w$, so $p_ k\sim w$. The $p_ k$ decrease as $k$ increases, and a $\theta $-plurisubharmonic limit would be a competitor for $P_\theta (v_1,\ldots ,v_ N)$. Suppose, to reach a contradiction, that $P_\theta (u_1,\ldots ,u_ N)\equiv -\infty $, so that $P_\theta (v_1,\ldots ,v_ N)\equiv -\infty $ as well. Then the $p_ k$ decrease to $-\infty $, and normalized compactness gives $\sup _ Xp_ k\to -\infty $.

Fix $s>0$ and choose $k>s$ so large that $p_ k\leq -s$. Iterating proposition 16.28.6, which bounds the measure of a rooftop by the measures of its obstacles on the contact sets, and using locality on $\{ v_ i>-s\} $, we obtain

Here we used $v_{i,k}\sim w$, and $w-k<-s$, which gives $v_{i,k}=v_ i$ on $\{ v_{i,k}>-s\} =\{ v_ i>-s\} $. Let $s\to \infty $. The measures $\mu _{v_ i}$ are finite and put no mass on pluripolar sets, so the resulting inequality contradicts the choice of the $v_ i$. $\square$