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Random sets without separability. (English) Zbl 0593.60018

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.

MSC:

60D05 Geometric probability and stochastic geometry
60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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