Proposition 22.1.37.21737B0303A5 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Then there exists a countable subfamily $I'\subseteq I$ such that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i=\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i. \]
Proof.
Fix a Kähler form $\omega _0$ on $X$. For each $m\geq 1$, proposition 17.15.31 gives a countable subfamily $I_ m\subseteq I$ such that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}=\operatorname *{sup}_{i\in I_ m}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}. \]
Set $I'=\bigcup _{m\geq 1}I_ m$. This is countable. We claim that it works.
Let $\omega '\in \textup{Käh}(X)$. Choose $m$ so large that $\omega '\geq m^{-1}\omega _0$. By proposition 22.1.36 and the compatibility of the transition maps,
\[ \begin{aligned} \operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +\omega '}[\Gamma ^ i]_{\mathcal{I}} =& P_{\theta +\omega '}\left[\operatorname *{sup}_{i\in I}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}\right]_{\mathcal{I}}\\ =& P_{\theta +\omega '}\left[\operatorname *{sup}_{i\in I'}{}\! {}^{*}P_{\theta +m^{-1}\omega _0}[\Gamma ^ i]_{\mathcal{I}}\right]_{\mathcal{I}}\\ =& \operatorname *{sup}_{i\in I'}{}\! {}^{*}P_{\theta +\omega '}[\Gamma ^ i]_{\mathcal{I}}. \end{aligned} \]
Thus the two non-Archimedean metrics have the same component in every class $\theta +\omega '$, and the assertion follows.
$\square$
Pome
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