0-1 laws for regular conditional distributions

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 $mathcal{A}$. Necessary and sufficient conditions for $mu(omega)(A)$ to be 0-1, for all $Ainmathcal{A}$ and $omegain A_0$, where $A_0inmathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $mathcal{A}$ is a tail sub-$sigma$-field. Let $H(omega)$ denote the $mathcal{A}$-atom including the point $omegainOmega$. Necessary and sufficient conditions for $mu(omega)(H(omega))$ to be 0-1, for all $omegain A_0$, are also given. If $(Omega,mathcal{B})$ is a standard space, the latter 0-1 law is true for various classically interesting sub-$sigma$-fields $mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$sigma$-fields connected with continuous time processes.