Proposition 22.1.5.B80F3474DB2F Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Then given $\omega ,\omega '\in \textup{Käh}(X)$ with $\omega \geq \omega '$, we have
\[ P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{-\infty } \right]=P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{-\infty } \right]_{\mathcal{I}} = \Gamma ^{\theta +\omega }_{-\infty }. \]
Proof.
Since for any $\tau <\Gamma _{\max }$, the potential $\Gamma ^{\theta +\omega '}_{\tau }$ is $\mathcal{I}$-good by example 18.1.11, it follows that
\[ P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{\tau } \right]=P_{\theta +\omega }\left[ \Gamma ^{\theta +\omega '}_{\tau } \right]_{\mathcal{I}}=\Gamma ^{\theta +\omega }_{\tau } \]
for all $\tau <\Gamma _{\max }$. Our assertion follows from proposition 16.4.9 and proposition 16.6.30.
$\square$
Pome
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