Theorem 16.28.700EA7F76611B5

Proof. After adding a constant to $u$, we may assume $u\leq \varphi $. Let $w\in \mathcal E_0$ be as in lemma 16.28.68. Since $w\in \mathcal E_0$, the envelope $P_\theta (w/\delta )$ is not identically $-\infty $, and $\delta P_\theta (w/\delta )\leq w$. The potential

belongs to $\mathrm{PSH}(X,(\varepsilon +\delta )\theta )$, so it is a competitor for $P_{(\varepsilon +\delta )\theta }(u)$. Applying lemma 16.28.69 with $a=\varepsilon +\delta $ and letting $\delta \downarrow 0$ gives the uniform estimate. The positivity follows from the same lemma with any one such $\delta $. $\square$