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October, 1975 On Existence and Non-Existence of Proper, Regular, Conditional Distributions
David Blackwell, Lester E. Dubins
Ann. Probab. 3(5): 741-752 (October, 1975). DOI: 10.1214/aop/1176996261

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 $\mathscr{B}$.

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David Blackwell. Lester E. Dubins. "On Existence and Non-Existence of Proper, Regular, Conditional Distributions." Ann. Probab. 3 (5) 741 - 752, October, 1975. https://doi.org/10.1214/aop/1176996261

Information

Published: October, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0348.60003
MathSciNet: MR400320
Digital Object Identifier: 10.1214/aop/1176996261

Subjects:
Primary: 60A05
Secondary: 60G05

Keywords: axiom of choice , Conditional distributions , normal conditional distributions , proper conditional distributions , Stochastic processes , stopping times

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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