Pome

Proposition 16.9.6C1D8A9BD3B3B

Proposition 16.9.6.C1D8A9BD3B3B Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then the following are equivalent:

  1. CCA0DECFB7E5

    $\varphi \sim _ P \psi $;

  2. A3A85D64324F

    $d_ S(\varphi ,\psi )=0$.

In particular, $d_ S(\varphi ,P_{\theta }[\varphi ])=0$ for all $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$.

Proof. By 16.8.13, we have $\varphi \sim _ P \psi $ if and only if $\varphi \sim _ P \varphi \lor \psi $ and $\psi \sim _ P \varphi \lor \psi $. By 16.9.4, $d_ S(\varphi ,\psi )=0$ if and only if $d_ S(\varphi ,\varphi \lor \psi )=0$ and $d_ S(\psi ,\varphi \lor \psi )=0$. So it suffices to prove the assertion when $\varphi \leq \psi $. Assuming this, by proposition 17.10.83, 2 holds if and only if

\[ \mathbf{E}\left(\ell ^{\varphi }\right)= \mathbf{E}\left(\ell ^{\psi }\right), \]

where $\mathbf{E}$ is introduced in definition 17.10.82. But by (17.10.94), this holds if and only if

\[ \sum _{j=0}^ n \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\sum _{j=0}^ n \int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \]

Thanks to the monotonicity theorem theorem 13.13.11, this holds if and only if for all $j=0,\ldots ,n$,

\[ \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j}, \]

which is equivalent to 1 by proposition 16.8.7. $\square$

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