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

Let $(Omega,mathcal{B},P)$ be a probability space, $mathcal{A}subsetmathcal{B}$ a sub-$sigma$-field, 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 $A_0inmathcal{A}$ such that $P(A_0)=1$ and $mu(omega)(A)in{0,1}$ for all $Ainmathcal{A}$ and $omegain A_0$. Provided $mathcal{B}$ is countably generated (and certain regular conditional distributions exist), the result applies whatever $P$ is.