PomeProposition 17.15.32.4D160D08BDC8 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Let $C\in \mathbb {R}$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} (\Gamma ^ i+C)=\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^ i+C. \]
Suppose that $(\Gamma '^ i)_{i\in I}$ is another family in $\mathrm{TC}(X,\theta ')_{>0}$ satisfying (17.15.21). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\leq \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma '^ i. \]
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.