Corollary 10.3.6134C98DC2772

Proof. Fix an index $i_0$. For $i\geq i_0$ the suprema are bounded above by $\sup _ X\varphi _{i_0}$, so the tail of the net lies in a compact subset of $L^1(X)$ by proposition 10.3.1, and a subnet converges in $L^1$ to some $\psi \in \mathrm{PSH}(X,\theta )$. For each fixed $i$, the members of the subnet are eventually below $\varphi _ i$, so $\psi \leq \varphi _ i$ almost everywhere, and therefore everywhere, as noted at the beginning of this section. Hence the pointwise infimum $\varphi $ is at least $\psi $ and is not identically $-\infty $. Applying theorem 5.10.1 on the coordinate balls of a finite cover, after adding local potentials of $\theta $, we find that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\varphi _ i\to \varphi $ in $L^1(X)$. The continuity of the supremum in proposition 10.3.1 gives the last assertion. $\square$