PomeProposition 16.5.4.557595391303 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then the following are equivalent:
88F668B67D98$\varphi \in \mathcal{E}(X,\theta ;\phi )$;
D4B2E9D2786A$P_{\theta }[\varphi ]=\phi $.
Proposition 16.5.4557595391303
Proof. 2 $\implies $ 1. Assume 2. By proposition 16.3.10,
\[ \int _ X\theta _\varphi ^ n=\int _ X\theta _{P_\theta [\varphi ]}^ n=\int _ X\theta _\phi ^ n. \]
Moreover, since the mass is positive, theorem 16.3.15 implies that
\[ \varphi \preceq P_\theta [\varphi ]=\phi . \]
Thus $\varphi \in \mathcal E(X,\theta ;\phi )$.
1 $\implies $ 2. Assume 1. Then $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Note that $\phi $ is a candidate of $P_{\theta }[\varphi ]$ as in (16.3.16). So $P_{\theta }[\varphi ]\geq \phi $. The reverse inequality is trivial. $\square$
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