## The Annals of Probability

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

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

David Blackwell and Lester E. Dubins

#### 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