Corollary 5.17.5B5E91AB35965

Proof. We may assume that $X$ is connected.

It suffices to show that $\varphi \not\equiv -\infty $. From this, it follows that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\varphi _ i\to \varphi $ in $L^1$ by proposition 5.14.2. The $L^1$-continuity of $\eta \mapsto \sup _ X\eta $ on $\mathrm{PSH}(X,\theta )$ from proposition 5.17.3 then gives the first equality in (5.17.6).

Fix $i_0\in I$ and replace $I$ by the cofinal tail $I_{\geq i_0}$. This does not change the infimum. On this tail we have

while the suprema are bounded from below by assumption. Hence proposition 5.17.3 shows that the net $(\varphi _ i)_{i\in I}$ is relatively compact with respect to the $L^1$-topology. We may therefore take a subnet $(\varphi _ j)_{j\in J}$ converging in $L^1$ to some $\psi \in \mathrm{PSH}(X,\theta )$. Since $(\varphi _ j)_{j\in J}$ is a Willard subnet of $(\varphi _ i)_{i\in I}$, we know that it is also decreasing. Therefore, $\psi =\inf _{j\in J} \varphi _ j$ due to proposition 5.14.2 and proposition 5.14.6.

Fix $i_0\in I$. For all sufficiently large $j\in J$, the corresponding index of the subnet dominates $i_0$, hence

Therefore, $\psi \leq \varphi _{i_0}$. As $i_0$ was arbitrary, $\psi \leq \varphi $ everywhere. In particular, $\varphi $ is not identically $-\infty $. $\square$