Corollary 16.9.12.77738CEC8071 Suppose that $(\varphi _ i)_{i\in I}$ is an increasing net in $\mathrm{PSH}(X,\theta )$, uniformly bounded from above. Then
\[ \varphi _ i\xrightarrow {d_ S} \operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i. \]
Proof.
Write $\varphi =\operatorname *{sup}_{j\in I}{}\! {}^{*}\varphi _ j$. Recall that by proposition 5.14.2, $\varphi \in \mathrm{PSH}(X,\theta )$. By 16.9.7, it suffices to show that for each $k=0,\ldots ,n$, we have
\[ \lim _{j\in I}\int _ X \theta _{\varphi _ j}^ k\wedge \theta _{V_{\theta }}^{n-k} = \int _ X \theta _{\varphi }^ k\wedge \theta _{V_{\theta }}^{n-k}. \]
The latter follows from corollary 13.13.12.
$\square$
Pome
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