A conditional 0-1 law for the symmetric sigma-field
Abstract
Let (,B,P) be a probability space, A a sub-sigma-field of B, and μ a regular conditional distribution for P given A. For various, classically interesting, choices of A (including tail and symmetric) the following 0-1 law is proved: There is a set A0 in A such that P(A0)=1 and μ(ω)(A) is 0 or 1 for all A in A and ω in A0. Provided B is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.
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