Proposition 17.15.28.C4A77DA880D9 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\in \mathrm{TC}(X,\theta )_{>0}$. Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then so is $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i$.
Finally, we have
3758587BAFAA
17.15.29
\begin{equation} \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)_{-\infty }=P_{\theta }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty } \right]. \end{equation}
Proof.
The first assertion and the second follow from proposition 17.15.22 and corollary 17.15.17.
It remains to argue (17.15.29). For this purpose, it suffices to show that
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)_{-\infty }\sim _ P \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty }. \]
For any $J\in \operatorname{Fin}(I)$, it follows from corollary 17.15.17 and proposition 16.8.28 that
\[ \left(\bigvee _{j\in J}\Gamma ^ j\right)_{-\infty }\sim _ P \bigvee _{j\in J}\Gamma ^ j_{-\infty }. \]
From this, applying proposition 16.4.9, proposition 16.8.28 and proposition 17.15.22, we conclude our assertion.
$\square$
Pome
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