Ross, David Random sets without separability. (English) Zbl 0593.60018 Ann. Probab. 14, 1064-1069 (1986). Summary: Suppose \({\mathcal V}\) and \({\mathcal F}\) are sets of subsets of X, for some fixed X. We apply König’s lemma from infinitary combinatorics to prove that if \({\mathcal V}\) and \({\mathcal F}\) satisfy some simple closure properties, and T is a Choquet capacity on X, then there is a probability measure on \({\mathcal F}\) such that for every \(V\in {\mathcal F}\), \(\{\) \(F\in {\mathcal F}:\) \(F\cap V\neq \emptyset \}\) is measurable with probability T(V). This extends the well-known case when \({\mathcal F}\) and \({\mathcal V}\) 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. Cited in 5 Documents MSC: 60D05 Geometric probability and stochastic geometry 60G57 Random measures 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) Keywords:random set; König’s lemma; infinitary combinatorics; Choquet capacity × Cite Format Result Cite Review PDF Full Text: DOI