Pome712499DFC37FCorollary 18.28.11.712499DFC37F Suppose that $f\colon K\rightarrow \mathbb {R}$ is bounded, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then there is a constant $C=C(\sup _ K f,K,X)>0$ such that
Proof. Since $K$ is non-pluripolar, 5.17.4 then gives a constant $C>0$, depending only on $\sup _ K f$, $K$ and $X$, such that every $\eta \in \mathrm{PSH}(X,\theta )$ satisfying $\eta |_ K\leq f$ quasi-everywhere satisfies $\eta \leq C$ on $X$.
This applies to the candidates defining both envelopes. Hence both candidate families are locally uniformly bounded above, so their upper semicontinuous suprema are genuine $\theta $-psh functions. By proposition 18.28.6 2, both envelopes are dominated by $f$ quasi-everywhere on $K$. Applying 5.17.4 once more gives
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