  Let $(\Omega,\mathcal{B},P)$ be a probability space,
$\mathcal{A}\subset\mathcal{B}$ 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 $A\in\mathcal{A}$ and
$\omega\in A_0$, where $A_0\in\mathcal{A}$ 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 $\omega\in\Omega$. Necessary and sufficient conditions for
$\mu(\omega)(H(\omega))$ to be 0--1, for all $\omega\in A_0$, 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.
