Proposition 5.14.3 (Choquet's lemma)`1D624ECF05E9`

Proposition 5.14.3 (Choquet's lemma).1D624ECF05E9 Assume that $X$ has countably many connected components. Assume that $(\varphi _ i)_{i\in I}$ is a non-empty family in $\mathrm{PSH}(X)$ that is locally uniformly bounded from above. There exists a countable subset $J\subseteq I$ such that

\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i=\operatorname *{sup}_{j\in J}{}\! {}^{*} \varphi _ j. \]

Proof. We may assume that $X$ is connected. Since by our convention, the complex manifold $X$ is paracompact, it can be covered by countably many holomorphic balls, so we can easily reduce to the case where $X$ is an open ball in $\mathbb {C}^ n$. In this case, the result is proved in [Lemma 4.31, GZ17]. $\square$

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@misc{pome-tag-1D624ECF05E9,
  author = {{Pome contributors}},
  title = {Proposition 5.14.3 (Choquet's lemma)},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/1D624ECF05E9/}
}

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Proposition 5.14.3 (Choquet's lemma)1D624ECF05E9

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Proposition 5.14.3 (Choquet's lemma)`1D624ECF05E9`