## The Annals of Probability

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

#### 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

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