0-1 laws for regular conditional distributions. (English) Zbl 1118.60004

Summary: 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.


60A10 Probabilistic measure theory
60A05 Axioms; other general questions in probability
60F20 Zero-one laws
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