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

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.