PomeCorollary 17.13.9.4B87EF03E3EA Let $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$. Then $\Gamma ^*_ t\in \mathcal{E}(X,\theta ;\phi )$ for all $t>0$.
Corollary 17.13.94B87EF03E3EA
Proof. Fix $t>0$. We already know that $\Gamma ^*_ t\in \mathrm{PSH}(X,\theta )$ by 17.13.7. It suffices to show that
\[ \Gamma ^*_ t\sim _ P \phi . \]
From (17.13.2) and proposition 16.8.28, we know that
\[ \Gamma ^*_ t\sim _ P \operatorname *{sup}_{\tau <\Gamma _{\max }}{}\! {}^{*} \Gamma _{\tau }=\phi . \]
This completes the proof. $\square$
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