A Conditional 0–1 Law for the Symmetric σ-field

Patrizia Berti () and Pietro Rigo ()
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Patrizia Berti: Universita’ di Modena e Reggio-Emilia
Pietro Rigo: Universita’ di Pavia

Journal of Theoretical Probability, 2008, vol. 21, issue 3, 517-526

Abstract: Abstract 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 $\mathcal{A}$ . For various, classically interesting, choices of $\mathcal{A}$ (including tail and symmetric), we prove the following 0–1 law: There is a set $A_{0}\in\mathcal{A}$ such that P(A 0)=1 and μ(ω)(A)∈{0,1} for all $A\in\mathcal{A}$ and ω∈A 0. If ℬ is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.

Keywords: 0–1 law; Invariant; tail and symmetric σ-fields; Regular conditional distribution; 60A05; 60A10; 60F20 (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s10959-008-0174-6

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