PomeProposition 21.13.57.BC781EB76285 Let $(\varphi _ i)_{i\in I}$ be a family in $\mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ uniformly bounded from above. Then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ and
\[ F_{\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i}=\bigvee _{i\in I}F_{\varphi _ i},\quad G_{\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i}=\operatorname{cl}\bigwedge _{i\in I} G_{\varphi _ i}. \]
Moreover, if $I$ is finite, then
\[ G_{\bigvee _{i\in I}\varphi _ i}=\bigwedge _{i\in I} G_{\varphi _ i}. \]
Similarly, if $\{ \varphi _ i\} _{i\in I}$ is a decreasing net in $\mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ such that $\inf _{i\in I}\varphi _ i\not\equiv -\infty $, then $\inf _{i\in I}\varphi _ i\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ and
\[ F_{\inf _{i\in I}\varphi _ i}=\bigwedge _{i\in I}F_{\varphi _ i},\quad G_{\inf _{i\in I}\varphi _ i}=\bigvee _{i\in I}G_{\varphi _ i}. \]
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