0-1 laws for regular conditional distributions

Let (ohm, B, P) be a probability space, A subset of B a sub-alpha-field, and mu a regular conditional distribution for P given A. Necessary and sufficient conditions for mu (w) (A) to be 0-1, for all A is an element of A and w is an element of A(0), where A(0) is an element of A and P(A(0)) = 1, are given. Such conditions apply, in particular, when A is a tail sub-alpha-field. Let H(w) denote the A-atom including the point w is an element of ohm. Necessary and sufficient conditions for mu(w) (H(w)) to be 0-1, for all w is an element of A(0), are also given. If (ohm, B) is a standard space, the latter 0-1 law is true for various classically interesting sub-alpha-fields A, including tail, symmetric, invariant, as well as some sub-alpha-fields connected with continuous time processes.