PomeProposition 16.4.3.8EB68D9BFB64 Let $(\varphi _ j)_{j\in I}$ be a decreasing net of potentials in $\mathrm{PSH}(X,\theta )$ satisfying $P_{\theta }[\varphi _ j]=\varphi _ j$ for each $j\in I$. Set $\varphi \coloneqq \inf _{j\in I} \varphi _ j$. Then $P_{\theta }[\varphi ]=\varphi $.
Proposition 16.4.38EB68D9BFB64
Proof. Since $P_{\theta }[\varphi _ j]=\varphi _ j$, we have $\sup _ X\varphi _ j=0$. By corollary 5.17.5, we have $\varphi \in \mathrm{PSH}(X,\theta )$, and $\varphi \leq 0$. Therefore, for each $j\in I$,
\[ \varphi \leq P_{\theta }[\varphi ]\leq P_{\theta }[\varphi _ j] =\varphi _ j. \]
Therefore, $\varphi =P_{\theta }[\varphi ]$. $\square$
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