The Annals of Probability

Random Sets Without Separability

David Ross

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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.

Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 1064-1069.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992459

Digital Object Identifier
doi:10.1214/aop/1176992459

Mathematical Reviews number (MathSciNet)
MR841605

Zentralblatt MATH identifier
0593.60018

JSTOR
links.jstor.org

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G57: Random measures 60G55: Point processes

Keywords
Random set Choquet capacity Konig's lemma

Citation

Ross, David. Random Sets Without Separability. Ann. Probab. 14 (1986), no. 3, 1064--1069. doi:10.1214/aop/1176992459. https://projecteuclid.org/euclid.aop/1176992459


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