Marginals and the product strong lifting problem
type: Item
instance of: scholarly article
title: Marginals and the product strong lifting problem
zbMATH Open document ID: 1440.28003
author: Maxim R. Burke
author: Nikolaos Demetrios Macheras
author: Werner Strauß
published in: Topology and its Applications
publication date: 2020-04-07T00:00:00Z
review text: In general a lifting problem asks for a map from a quotient structure back to the original structure whose composition with the quotient map is the identity. The present paper deals with liftings for topological probability measure spaces. If \((X,\Sigma, \mathcal{T},\mu)\) is such a space then the quotient is usually modulo the ideal \(\mathcal{N}\) of \(\mu\)-null sets. A lifting \(\lambda:\Sigma/\mathcal{N}\to\Sigma\) usually replaced by \(\sigma=\lambda\circ q\), where \(q:\Sigma\to\Sigma/\mathcal{N}\) is the quotient map. The map~\(\sigma:\Sigma\to\Sigma\) has two properties: 1)~\(A=_\mu\sigma(A)\), and 2)~``if \(A=_\mu B\) then \(\sigma(A)=\sigma(B)\)''. Quite often a map like~\(\sigma\) is itself already called a lifting and in case \(\lambda\)~is a homomorphism then \(\sigma\) is a homomorphism `modulo null-sets'. The main result of the paper is an improvement of Theorem~4.3 of [\textit{K. Musiał} et al., J. Theor. Probab. 20, No. 3, 545--560 (2007; Zbl 1143.28003)] concerning liftings for product spaces, given liftings for the factors. Unfortunately the detail in the assumptions and conclusions make it impracticable give a summary that would do the theorem justice; we refer the interested reader to the paper itself for more information.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 28A51
Mathematics Subject Classification ID: 28A35
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 7187668
zbMATH Keywords: topological measure space
zbMATH Keywords: lifting problem
zbMATH Keywords: products
MaRDI profile type: Publication
full work available at URL: j.topol.2019.107021
OpenAlex ID: W2996471535
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DOI: 10.1016/J.TOPOL.2019.107021
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description: scientific article; zbMATH DE number 7187668