Lemma 17.15.24.0B89BAEA6D39 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right]= \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]. \]
Proof.
Observe that
\[ \left( P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right] \right)_{\max }= \left(\operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]\right)_{\max }=\sup _{i\in I}\Gamma ^ i_{\max }. \]
Fix $\tau \in \mathbb {R}$ less than this common value.
It suffices to show that
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\tau }\sim _ P \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[\Gamma ^{i}\right] \right)_{\tau }. \]
This is an immediate consequence of proposition 16.8.28.
$\square$
Pome
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