The Annals of Probability

On Existence and Non-Existence of Proper, Regular, Conditional Distributions

David Blackwell and Lester E. Dubins

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Abstract

If $\mathscr{A}$ is the tail, invariant, or symmetric field for discrete-time processes, or a field of the form $\mathscr{F}_{t+}$ for continuous-time processes, then no countably additive, regular, conditional distribution given $\mathscr{A}$ is proper. A notion of normal conditional distributions is given, and there always exist countably additive normal conditional distributions if $\mathscr{A}$ is a countably generated sub $\sigma$-field of a standard space. The study incidentally shows that the Borel-measurable axiom of choice is false. Classically interesting subfields $\mathscr{A}$ of $\mathscr{B}$ possess certain desirable properties which are the defining properties for $\mathscr{A}$ to be "regular" in $\mathscr{B}$.

Article information

Source
Ann. Probab. Volume 3, Number 5 (1975), 741-752.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996261

Digital Object Identifier
doi:10.1214/aop/1176996261

Mathematical Reviews number (MathSciNet)
MR400320

Zentralblatt MATH identifier
0348.60003

JSTOR
links.jstor.org

Subjects
Primary: 60A05: Axioms; other general questions
Secondary: 60G05: Foundations of stochastic processes

Keywords
Conditional distributions proper conditional distributions normal conditional distributions stochastic processes axiom of choice stopping times

Citation

Blackwell, David; Dubins, Lester E. On Existence and Non-Existence of Proper, Regular, Conditional Distributions. Ann. Probab. 3 (1975), no. 5, 741--752. doi:10.1214/aop/1176996261. https://projecteuclid.org/euclid.aop/1176996261


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