PomeTheorem 16.9.49.88D575A3C46A The map
\[ P_{\theta }[\bullet ]_{\mathcal{I}}\colon \mathrm{PSH}(X,\theta )_{>0}\rightarrow \mathrm{PSH}(X,\theta )_{>0} \]
is continuous with respect to $d_ S$.
Theorem 16.9.4988D575A3C46A
Proof. Let $(\varphi _ i)_{i\in \mathbb {Z}_{>0}}$ be a sequence in $\mathrm{PSH}(X,\theta )_{>0}$ such that $\varphi _ i\xrightarrow {d_ S}\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. We want to show that
6.73
\begin{equation} P_{\theta }[\varphi _ i]_{\mathcal{I}}\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}
We may assume that the $\varphi _ i$'s and $\varphi $ are all model potentials by proposition 16.9.6.
By proposition 16.9.17 and corollary 16.9.39, we may assume that $(\varphi _ i)_ i$ is either increasing or decreasing. In the increasing case, we apply proposition 16.6.30 and corollary 16.9.12, while in the decreasing case, we apply proposition 16.6.29, proposition 16.4.4 and corollary 16.9.14. $\square$
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