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

TL;DR

This paper proves a 0-1 law for symmetric sigma-fields, showing that under certain conditions, the regular conditional distribution is almost surely degenerate on these fields, extending classical results.

Contribution

It establishes a new conditional 0-1 law for symmetric sigma-fields applicable to any probability measure with countably generated sigma-algebra.

Findings

Existence of a set in the sigma-field with probability one where the conditional distribution is degenerate.

The law applies regardless of the underlying probability measure, given certain regularity conditions.

The result generalizes classical 0-1 laws to symmetric and tail sigma-fields.

Abstract

Let (\Omega,\mathcal{B},P) be a probability space, \mathcal{A} a sub-sigma-field of \mathcal{B}, and \mu a regular conditional distribution for P given \mathcal{A}. For various, classically interesting, choices of \mathcal{A} (including tail and symmetric) the following 0-1 law is proved: There is a set A_0 in \mathcal{A} such that P(A_0)=1 and \mu(\omega)(A) is 0 or 1 for all A in \mathcal{A} and \omega in A_0. Provided \mathcal{B} is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.