Proposition 16.6.11363E28FF4A61

Proof. In view of proposition 16.6.2, it suffices to show that for $k \in \mathbb {Z}_{>0}$, we have

By proposition 5.14.3, we can find $\psi _ i\in \mathrm{PSH}(X,\theta )$ ($i\in \mathbb {Z}_{>0}$) such that $\psi _ i\leq 0$, $\psi _ i\sim _{\mathcal{I}}\varphi $ for all $i\geq 1$ and

By proposition 16.6.5, we may replace $\psi _ i$ by $\psi _1\lor \cdots \lor \psi _ i$ and assume that the sequence $\psi _ i$ is increasing. In this case, it follows from the strong openness theorem theorem 5.16.21 that for each $k\in \mathbb {Z}_{>0}$, we have

for $j$ large enough. $\square$