Let (Ω, ℬ, P) be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-σ-field, and μ a regular conditional distribution for P given $\mathcal{A}$ . Necessary and sufficient conditions for μ(ω)(A) to be 0–1, for all $A\in\mathcal{A}$ and ω∈A0, where $A_{0}\in\mathcal{A}$ and P(A0)=1, are given. Such conditions apply, in particular, when $\mathcal{A}$ is a tail sub-σ-field. Let H(ω) denote the $\mathcal{A}$ -atom including the point ω∈Ω. Necessary and sufficient conditions for μ(ω)(H(ω)) to be 0–1, for all ω∈A0, are also given. If (Ω, ℬ) is a standard space, the latter 0–1 law is true for various classically interesting sub-σ-fields $\mathcal{A}$ , including tail, symmetric, invariant, as well as some sub-σ-fields connected with continuous time processes.