Theorem 16.9.57.CAB6F4E23080 Let $(\varphi _ j)_{j\in I}$ be a net in $\mathrm{PSH}(X,\theta )$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that $\varphi _ j\xrightarrow {d_ S}\varphi $. Then
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16.9.58
\begin{equation} \operatorname{vol}\theta _{\varphi _ j}\to \operatorname{vol}\theta _{\varphi },\quad \int _ X \theta _{\varphi _ j}^ n\to \int _ X \theta _{\varphi }^ n. \end{equation}
Proof.
The latter part of (16.9.58) is just a special case of theorem 16.9.30. It remains to prove the former part.
We may therefore assume that $\int _ X \theta _{\varphi _ j}^ n>0$ for all $j\in I$ after passing to a tail. Then by theorem 16.9.49, we have
\[ P_{\theta }[\varphi _ j]_{\mathcal{I}}\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}. \]
Therefore, the first part of (16.9.58) follows again from theorem 16.9.30.
$\square$
Pome
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