Definition 17.15.204CC845CC4171

Definition 17.15.20.4CC845CC4171 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$. Assume that

Then we define $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}\in \mathrm{TC}(X,\theta )_{>0}$ as follows:

1. E265E4ED4EFB

   We set

   \[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\max }=\sup _{i\in I}\Gamma ^{i}_{\max }; \]
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   for any $\tau < \sup _{i\in I}\Gamma ^{i}_{\max }$, we let

   \[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\tau }\coloneqq \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}_{\tau }. \]