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March 2007 0–1 laws for regular conditional distributions
Patrizia Berti, Pietro Rigo
Ann. Probab. 35(2): 649-662 (March 2007). DOI: 10.1214/009117906000000845


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.


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Patrizia Berti. Pietro Rigo. "0–1 laws for regular conditional distributions." Ann. Probab. 35 (2) 649 - 662, March 2007.


Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1118.60004
MathSciNet: MR2308591
Digital Object Identifier: 10.1214/009117906000000845

Primary: 60A05 , 60A10 , 60F20

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

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 2 • March 2007
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