Proposition 16.4.93654A10A0CCC

Proof. We may assume that $I$ is infinite since otherwise, there is nothing to prove.

We write

Then it is clear that $\eta \leq P_{\theta }[\varphi ]$.

By corollary 13.13.12, we have

We may assume that for each $i\in I$, $\int _ X\theta _{\varphi _ i}^ n<\int _ X\theta _{\varphi }^ n$, since otherwise $P_{\theta }[\varphi _ i]=P_{\theta }[\varphi ]$ by theorem 16.3.15, and the same holds for all $j\geq i$; in that case there is nothing to prove.

So by 13.13.30, we can find a decreasing net $\epsilon _ i\searrow 0$ ($i\in I$) with $\epsilon _ i\in (0,1)$ and $\psi _ i\in \mathrm{PSH}(X,\theta )$ ($i\in I$) such that for all $i\in I$,

We may, for example, take

By the concavity property of the envelope proposition 16.4.2, we have

Each $P_{\theta }[\psi _ i]$ has supremum $0$, hence the family is bounded in $L^1$ by proposition 5.17.3. Thus $\epsilon _ iP_{\theta }[\psi _ i]\to 0$ in $L^1$. The previous inequality then gives $P_{\theta }[\varphi ]\leq \eta $ almost everywhere, and hence everywhere by proposition 5.14.6. $\square$