Isomorphisms for convergence structures. II: Borel spaces and infinite measures
type: Item
instance of: scholarly article
title: Isomorphisms for convergence structures. II: Borel spaces and infinite measures
zbMATH Open document ID: 0906.28003
author: Yeneng Sun
published in: Advances in Mathematics
publication date: 1999-01-25T00:00:00Z
review text: This paper is a continuation of an earlier author's paper [[1] Adv. Math. 116, No. 2, 322-355 (1995; Zbl 0867.28003)], in which a new notion of isomorphism is introduced. In [1] a construction of such isomorphism from a Polish space with nonatomic Borel measure to the unit interval with Lebesgue measure is given. In the present paper the considerations from [1] are extended to general Borel spaces (instead of Polish spaces). A topological space is called a Borel space if it is homeomorphic to a Borel subset of a complete separable metric space. Let \((X,\mu)\), \((Y,\nu)\) be two topological measure spaces. A Borel measurable, measure-preserving mapping \(F:X\to Y\) is called an isomorphism if \(F\) is \(\mu\)-continuous and \(F^{-1}\) is a \(\nu\)-continuous mapping. If such an isomorphism \(F: X\to Y\) exists, then \((X,\mu)\) is said to be isomorphic to \((Y,\nu)\). A typical result: Let \(\nu\) be a finite positive measure on a Borel space \(X\). Put \(\nu(X)= b\). Then the following two assertions are equivalent: (a) There exists a measure \(\mu\) on \([0, b]\) with \(\mu([0, b])= b\) such that \((X,\nu)\) is isomorphic to \(([0,b],\mu)\). (b) There is a Polish space \(Y\) embedded in \(X\) such that \(\nu(X\setminus Y)= 0\) and every atom of \(\nu\) is a condensation point of \(X\).
reviewed by: Tibor Šalát
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1006700
zbMATH Keywords: isomorphisms
zbMATH Keywords: convergence
zbMATH Keywords: Borel spaces
zbMATH Keywords: Polish spaces
MaRDI profile type: Publication
full work available at URL: aima.1996.0078
OpenAlex ID: W69545987
DOI: 10.1006/AIMA.1996.0078
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description: scientific article; zbMATH DE number 1006700