Lemma 16.9.27.51EA136E15FA Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$ and $\varphi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi _ i\xrightarrow {d_ S}\varphi $. Then for any $t\in (0,1]$,
\[ (1-t)\varphi _ i+ tV_{\theta }\xrightarrow {d_ S}(1-t)\varphi + tV_{\theta }. \]
Proof.
Fix $t\in (0,1]$. We write
\[ \varphi _{i,t}=(1-t)\varphi _ i+ tV_{\theta },\quad \varphi _ t=(1-t)\varphi + tV_{\theta } \]
for any $i\in I$.
By corollary 16.9.9, it suffices to show that for each $j=0,\ldots ,n$,
D81A5AAFDA65
16.9.28
\begin{equation} 2\int _ X \theta _{\varphi _{i,t}\lor \varphi _ t}^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi _{i,t}}^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi _ t}^ j\wedge \theta _{V_{\theta }}^{n-j}\to 0. \end{equation}
Observe that
\[ \varphi _{i,t}\lor \varphi _ t=(1-t)(\varphi \lor \varphi _ i)+t V_{\theta }. \]
So after binomial expansion, (16.9.28) follows from corollary 16.9.9.
$\square$
Pome
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