Abstract
Suppose $\mathscr{V}$ and $\mathscr{F}$ are sets of subsets of $X$, for some fixed $X$. We apply Konig's lemma from infinitary combinatorics to prove that if $\mathscr{V}$ and $\mathscr{F}$ satisfy some simple closure properties, and $T$ is a Choquet capacity on $X$, then there is a probability measure on $\mathscr{F}$ such that for every $V \in \mathscr{F}, \{F \in \mathscr{F}: F \cap V \neq \varnothing\}$ is measurable with probability $T(V)$. This extends the well-known case when $\mathscr{F}$ and $\mathscr{V}$ are the closed (respectively, open) subsets of a second countable Hausdorff space $X$. The result enables us to define a general notion of "random measurable set"; for example, we can build a point process with Poisson distribution on any infinite (possibly nontopological) measure space.
Citation
David Ross. "Random Sets Without Separability." Ann. Probab. 14 (3) 1064 - 1069, July, 1986. https://doi.org/10.1214/aop/1176992459
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