Products of Blackwell spaces and regular conditional probabilities
type: Item
instance of: scholarly article
title: Products of Blackwell spaces and regular conditional probabilities
zbMATH Open document ID: 0581.28002
author: R. M. Shortt
published in: Real Analysis Exchange
publication date: 1985-01-01T00:00:00Z
review text: [Cf. the review above for notations and definitions.] - In previous work, the author investigated the Blackwell property for sets \(X\subseteq {\mathbb{R}}\) such that \({\mathbb{R}}\setminus X\) is totally imperfect (contains no Cantor-like sets). Here, the following is proved: Proposition: Let \(X\subseteq {\mathbb{R}}\) have totally imperfect complement. Let A be an analytic set. For each \(n\geq 1\), the following are equivalent: 1) \(X^ n\) has the strong Blackwell property; 2) \(X^ n\) has the Blackwell property; 3) \(X^ n\times {\mathbb{R}}\) has the strong Blackwell property; 4) \(X^ n\times {\mathbb{R}}\) has the Blackwell property; 5) \(X^ n\times A\) has the strong Blackwell property; 6) \(X^ n\times A\) has the Blackwell property. An example is given of a probability space whose Borel structure is strong Blackwell but for which regular conditional probabilities do not always exist. This answers a question of D. Ramachandran.
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 3929351
zbMATH Keywords: Borel \(\sigma \) -algebra
zbMATH Keywords: separable metric space
zbMATH Keywords: Blackwell property
zbMATH Keywords: analytic set
zbMATH Keywords: regular conditional probabilities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3929351