0--1 laws for regular conditional distributions

Abstract

Let $(\Omega ,\mathcal{B},P)$ be a probability space, $\mathcal{A}\subset \mathcal{B}$ a sub-$\sigma$-field, and $\mu$ a regular conditional distribution for $P$ given $\mathcal{A}$. Necessary and sufficient conditions for $\mu (\omega )(A)$ to be 0--1, for all $A\in \mathcal{A}$ and $\omega \in A_0$, where $A_0\in \mathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathcal{A}$ is a tail sub-$\sigma$-field. Let $H(\omega )$ denote the $\mathcal{A}$-atom including the point $\omega \in \Omega$. Necessary and sufficient conditions for $\mu (\omega )(H(\omega ))$ to be 0--1, for all $\omega \in A_0$, are also given. If $(\Omega ,\mathcal{B})$ is a standard space, the latter 0--1 law is true for various classically interesting sub-$\sigma$-fields $\mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$\sigma$-fields connected with continuous time processes.

Published at http://dx.doi.org/10.1214/009117906000000845 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (1)

• A conditional 0-1 law for the symmetric sigma-field