Pome

Proposition 16.9.47A2F662346EB5

Proposition 16.9.47.A2F662346EB5 Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$, decreasing with respect to the $P$-partial order. Assume that there is $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ with $\varphi _ i\xrightarrow {d_ S}\varphi $, then

17876C1905F0 16.9.48
\begin{equation} \inf _{i\in I} P_{\theta }[\varphi _ i]=P_{\theta }[\varphi ]. \end{equation}

Proof. First observe that by our assumption, $(P_{\theta }[\varphi _ i])_{i\in I}$ is a decreasing net. By the continuity of non-pluripolar masses corollary 16.9.13, we have

\[ \lim _{i\in I} \int _ X \theta _{P_{\theta }[\varphi _ i]}^ n=\lim _{i\in I} \int _ X \theta _{\varphi _ i}^ n= \int _ X \theta _{\varphi }^ n. \]

Replacing $I$ by a tail, we may therefore assume that $\varphi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for each $i\in I$. Since $\varphi _ i\sim _ P P_\theta [\varphi _ i]$, replacing $\varphi _ i$ by $P_\theta [\varphi _ i]$ does not change its $d_ S$-class by proposition 16.9.6. We may assume that each $\varphi _ i$ is model and $(\varphi _ i)_{i\in I}$ is decreasing. In this case, by proposition 16.4.4, the potential

\[ \psi \coloneqq \inf _{i\in I} \varphi _ i \]

is model in $\mathrm{PSH}(X,\theta )_{>0}$ with

\[ \int _ X \theta _{\psi }^ n=\lim _{i\in I} \int _ X \theta _{\varphi _ i}^ n= \int _ X \theta _{\varphi }^ n. \]

Therefore, by corollary 16.9.14, we find $\varphi _ i\xrightarrow {d_ S}\psi $. In particular, $\varphi \sim _ P \psi $ by proposition 16.9.6. Therefore,

\[ \psi =P_{\theta }[\psi ]=P_{\theta }[\varphi ]. \]

Therefore, (16.9.48) follows. $\square$

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