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

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata “G. Vitali”, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100, Modena, Italy

    Patrizia Berti

  2. Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100, Pavia, Italy

    Pietro Rigo

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  1. Patrizia Berti
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  2. Pietro Rigo
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Correspondence to Pietro Rigo.

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Berti, P., Rigo, P. A Conditional 0–1 Law for the Symmetric σ-field. J Theor Probab 21, 517–526 (2008). https://doi.org/10.1007/s10959-008-0174-6

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  • DOI: https://doi.org/10.1007/s10959-008-0174-6

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