The Annals of Probability

0–1 laws for regular conditional distributions

Patrizia Berti and Pietro Rigo

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Abstract

Let $(Ω, ℬ, P)$ be a probability space, $\mathscr{A}\subset\mathscr{B}$ a sub-$σ$-field, and $μ$ a regular conditional distribution for $P$ given $\mathscr{A}$. Necessary and sufficient conditions for $μ(ω)(A)$ to be 0–1, for all $A\in\mathscr{A}$ and $ω∈A_0$, where $A_{0}\in\mathscr{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathscr{A}$ is a tail sub-$σ$-field. Let $H(ω)$ denote the $\mathscr{A}$-atom including the point $ω∈Ω$. Necessary and sufficient conditions for $μ(ω)(H(ω))$ to be 0–1, for all $ω∈A_0$, are also given. If $(Ω, ℬ)$ is a standard space, the latter 0–1 law is true for various classically interesting sub-$σ$-fields $\mathscr{A}$, including tail, symmetric, invariant, as well as some sub-$σ$-fields connected with continuous time processes.

Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 649-662.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1175287757

Digital Object Identifier
doi:10.1214/009117906000000845

Mathematical Reviews number (MathSciNet)
MR2308591

Zentralblatt MATH identifier
1118.60004

Subjects
Primary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60F20: Zero-one laws

Keywords
0–1 law measurability regular conditional distribution tail σ-field

Citation

Berti, Patrizia; Rigo, Pietro. 0–1 laws for regular conditional distributions. Ann. Probab. 35 (2007), no. 2, 649--662. doi:10.1214/009117906000000845. https://projecteuclid.org/euclid.aop/1175287757


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