On piecewise injective measurable maps
type: Item
instance of: scholarly article
title: On piecewise injective measurable maps
zbMATH Open document ID: 1186.28003
author: A. V. Chistyakov
published in: Russian Mathematics
publication date: 2009-06-10T00:00:00Z
review text: Let \(X\) be a Polish space and let \(\mu\) be the completion of a positive finite Borel measure on \(X\). Denote by \({\mathcal B}_\mu(X)\) the domain of \(\mu\). The author calls a \(\mu\)-measurable map \(\alpha: X\to Y\), where \(Y\) is another Polish space, almost piecewise injective if there exists a countable family \(\{X_i: i\in I\}\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\bigcup_{i\in I}X_i)= 0\) and \(\alpha|X_i\) is injective for all \(i\in I\). It is shown that this condition is equivalent to each of the following ones: (1) there exists \(\widetilde X\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\widetilde X)= 0\) and \(\alpha^{-1}(y)\cap\widetilde X\) is countable for all \(y\in Y\); (2) there exists \(\widetilde X\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\widetilde X)= 0\) and for every \(A\subset\widetilde X\) with \(\mu(A)= 0\) we have \(\mu(\alpha^{-1}(\alpha(A)))= 0\). The proofs are based on some classical results from the descriptive theory of Polish spaces mainly due to Luzin and Mazurkiewicz. Unfortunately, the English of the paper is rather bad and some proofs are lengthy.
reviewed by: Zbigniew Lipecki
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28C15
zbMATH DE Number: 5563683
zbMATH Keywords: Polish space
zbMATH Keywords: Borel measure
zbMATH Keywords: measurable map
zbMATH Keywords: piecewise injective map
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5563683