Open Access
July, 1986 Random Sets Without Separability
David Ross
Ann. Probab. 14(3): 1064-1069 (July, 1986). DOI: 10.1214/aop/1176992459


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.


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David Ross. "Random Sets Without Separability." Ann. Probab. 14 (3) 1064 - 1069, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60018
MathSciNet: MR841605
Digital Object Identifier: 10.1214/aop/1176992459

Primary: 60D05
Secondary: 60G55 , 60G57

Keywords: Choquet capacity , Konig's lemma , random set

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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