PomeA1C36A6249C7Proposition 18.28.12.A1C36A6249C7 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Suppose that $\varphi \preceq _{P} \psi $. Then
Proof. Let $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ be such that $\eta |_ K\leq f$ quasi-everywhere and $\eta \preceq \varphi $. Then $\eta \preceq _ P \psi $, and thanks to corollary 16.8.30 for any $C>0$, we have $\eta \land (\psi +C)\in \mathrm{PSH}(X,\theta )_{>0}$. But then $\eta \land (\psi +C)$ is a candidate in the definition of $P_{\theta ,K}[\psi ](f)$, and we find
By corollary 16.8.30 again, we get
Our assertion follows now from 18.28.7. $\square$
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