0--1 laws for regular conditional distributions

Abstract

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