PomeCorollary 16.9.9.B29DCA70A9E6 For any $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$, we have
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16.9.10
\begin{equation} \begin{split} d_ S(\varphi ,\psi )\leq & \frac{1}{n+1}\sum _{j=0}^ n \left( 2\int _ X \theta _{\varphi \lor \psi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j} \right) \\ \leq & C_ n d_ S(\varphi ,\psi ), \end{split} \end{equation}
where $C_ n=3(n+1)2^{n+2}$.
In particular, if $(\varphi _ i)_{i\in I}$ is a net in $\mathrm{PSH}(X,\theta )$ with $d_ S$-limit $\varphi $, then for each $j=0,\ldots ,n$,
\[ \lim _{i\in I}\int _ X \theta _{\varphi _ i}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\lim _{i\in I}\int _ X \theta _{\varphi _ i\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \]
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