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Acta Mathematica Vietnamica

CHU SPACES AND CONDITIONAL PROBABILITY

NGUYEN NHUY, NGUYEN VAN QUANG

Abstract

Let Ω~=(Ω,P,A) and Σ~=(Σ,Q,B) be probability measure spaces, and let ϕ:ΩΣ and ψ:ΣΩ be measurability preserving maps. The maps ϕ and ψ induce ϕ1:BA and ψ1:AB. By (A,A,f) we denote the Chu space associated with the probability measure space (Ω,P,A). Our main results are:

Theorem 1. Let P(Ω~)=(A,A,f) and P(Σ~)=(B,B,g) be Chu spaces associated with Ω~ and Σ~, respectively. If Φ=(ψ1,ϕ1):P(Ω~)P(Σ~) is a Chu morphism, then both ϕ and ψ are measure preserving.

Theorem 2. The pair (ϕ,ψ) is mutually measure preserving if and only if Φ=(ψ1,ϕ1):(A,A,f)(B,B,g) is a Chu morphism.