Pome

Proposition 17.15.314B67ABC0083D

Proposition 17.15.31.4B67ABC0083D Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then there is a countable subset $I'\subseteq I$ such that

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i =\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i. \]

Proof. We may assume that $I$ is infinite.

Set

\[ M\coloneqq \sup _{i\in I}\Gamma ^ i_{\max }. \]

For each rational number $q<M$, apply Choquet's lemma proposition 5.14.3 to the locally uniformly bounded above family

\[ \left\{ \left(\bigvee _{j\in J}\Gamma ^ j\right)_ q:J\in \operatorname{Fin}(I)\right\} \subseteq \mathrm{PSH}(X,\theta ). \]

We obtain a countable subfamily $\mathcal J_ q\subseteq \operatorname{Fin}(I)$ such that

\[ \operatorname *{sup}_{J\in \operatorname{Fin}(I)}{}\! {}^{*}\left(\bigvee _{j\in J}\Gamma ^ j\right)_ q = \operatorname *{sup}_{J\in \mathcal J_ q}{}\! {}^{*}\left(\bigvee _{j\in J}\Gamma ^ j\right)_ q. \]

Let

\[ I'\coloneqq \bigcup _{q\in \mathbb Q, q<M} \bigcup _{J\in \mathcal J_ q}J. \]

Then $I'$ is countable and, for every rational $q<M$,

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

Moreover, $\sup _{i\in I'}\Gamma ^ i_{\max }=M$, since the last equality is non-trivial for every rational $q<M$. Let $\Gamma '=\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i$. Then clearly, $\Gamma '\leq \Gamma $. We claim that they are actually equal.

Thanks to proposition 16.8.7 and 17.12.6, it suffices to show that for any $\tau <\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{\max }$, we have

\[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma '_{\tau }\right)^ n=\int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau }\right)^ n. \]

The equality is already known for all rational $\tau <M$. Since both

\[ \tau \mapsto \log \int _ X\theta _{\Gamma '_\tau }^ n,\quad \tau \mapsto \log \int _ X\theta _{\Gamma _\tau }^ n \]

are continuous on $(-\infty ,M)$ by proposition 17.12.9, the equality of masses holds for every $\tau <M$. $\square$

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