Lemma 16.9.7.03FCA1AB2751 Suppose that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ and $\varphi \preceq _ P \psi $. Then
\[ d_ S(\varphi ,\psi )=\frac{1}{n+1}\sum _{j=0}^ n \left(\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\right). \]
Proof.
Set $\chi \coloneqq \varphi \lor \psi $. By 16.8.13, we have $\chi \sim _ P\psi $. Hence proposition 16.9.6 gives $d_ S(\chi ,\psi )=0$, and the pseudometric property gives
\[ d_ S(\varphi ,\psi )=d_ S(\varphi ,\chi ). \]
Since $\varphi \leq \chi $, (17.10.94) and proposition 17.10.83 give
\[ d_ S(\varphi ,\chi )=\frac{1}{n+1}\sum _{j=0}^ n \left(\int _ X \theta _{\chi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\right). \]
Finally, proposition 16.8.7 applied to $\psi \leq \chi $ gives
\[ \int _ X \theta _{\chi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j} \]
for all $j=0,\ldots ,n$.
$\square$
Pome
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