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Corollary 18.28.249790F808BF37

Corollary 18.28.24.9790F808BF37 Let $v\in C^0(K)$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

FFCF7AC9E703 18.28.25
\begin{equation} \begin{aligned} \int _{(X\setminus K)\cup \left\{ P_{\theta ,K}[\varphi ](v)<v\right\} }\theta ^ n_{P_{\theta ,K}[\varphi ](v)}=& 0,\\ \int _{(X\setminus K)\cup \left\{ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)<v\right\} }\theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}=& 0. \end{aligned} \end{equation}

Proof. The first equation in (18.28.25) follows from 18.28.21. For the second, it suffices to observe that

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)=P_{\theta ,K}\left[P_{\theta }[\varphi ]_{\mathcal{I}}\right]_{\mathcal{I}}(v)=P_{\theta ,K}\left[P_{\theta }[\varphi ]_{\mathcal{I}}\right](v) \]

by corollary 18.28.14. $\square$

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