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

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