0–1 laws for regular conditional distributions

Abstract

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