PomeProposition 18.1.19.FD679E0ECD3D Let $(\varphi _ j)_{j\in I}$ be a non-empty family of $\mathcal{I}$-good potentials in $\mathrm{PSH}(X,\theta )$ for some closed real smooth $(1,1)$-form $\theta $ on $X$. Assume that this family is locally uniformly bounded from above. Then $\operatorname *{sup}_{j\in I}{}\! {}^{*} \varphi _ j$ is $\mathcal{I}$-good.
Proposition 18.1.19FD679E0ECD3D
Proof. After adding a Kähler form to $\theta $, we may assume that $\varphi _ j\in \mathrm{PSH}(X,\theta )_{>0}$ for all $j\in I$.
When $I$ is finite, this result follows from proposition 18.1.17. When $I$ is infinite, by Choquet's lemma proposition 5.14.3, we may assume that $I=\mathbb {Z}_{>0}$. Replacing $\varphi _ j$ by $\varphi _1\lor \cdots \lor \varphi _ j$ and using proposition 18.1.17, we may assume that the sequence $(\varphi _ j)_ j$ is increasing. In this case, as shown in corollary 16.9.12,
\[ \varphi _ j\xrightarrow {d_ S}\operatorname *{sup}_{ i\in \mathbb {Z}_{>0}}{}\! {}^{*} \varphi _ i. \]
Therefore, $\operatorname *{sup}_{i\in \mathbb {Z}_{>0}}{}\! {}^{*} \varphi _ i$ is $\mathcal{I}$-good by corollary 18.1.9. $\square$
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