Isomorphisms for convergence structures
type: Item
instance of: scholarly article
title: Isomorphisms for convergence structures
zbMATH Open document ID: 0867.28003
author: Yeneng Sun
published in: Advances in Mathematics
publication date: 1997-07-06T00:00:00Z
review text: The paper is a continuation of ideas of Kuratowski and von Neumann [cf. \textit{J. von Neumann}, Ann. Math., II. Ser. 33, 574-586 (1932; Zbl 0005.05603)]. Let \((X,\mu)\), \((Y,\nu)\) be topological measure spaces and let \(F:X\to Y\) be a Borel measurable, measure-preserving, bijective mapping. The function \(F\) is said to be an isomorphism provided that \(F\) is \(\mu\)-continuous and \(F^{-1}\) is \(\nu\)-continuous. The space \((X,\mu)\) is said to be isomorphic to \((Y,\nu)\) if there exists an isomorphism \(F:X\to Y\). The main result: Let \(\mu\) be a non-atomic Borel probability measure on the Polish space \(X\) (\(X\) is homeomorphic to a complete separable metric space). Let \(D\) be a countable set in the support of \(\mu\). Then there exists an isomorphism \(F\) from \((X,\mu)\) to \((I,\lambda)\) (\(I\) is the unit interval, \(\lambda\) is the Lebesgue measure) such that \(F\) is continuous at every point in \(D\) and \(F^{-1}\) is continuous at every point in \(F(D)\). Corollary: Let \((X,\mu)\) have the previous meaning. Then there exists a subset \(Y\) of \(I\) and a Borel probability measure \(\nu\) on \(Y\) such that \((X,\mu)\) is isomorphic to \((Y,\nu)\). Many examples and applications are given mainly in general theory of uniform distribution of sequences.
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 11K36
zbMATH DE Number: 863625
zbMATH Keywords: Borel measurable bijective mapping
zbMATH Keywords: measure-preserving mapping
zbMATH Keywords: convergence
zbMATH Keywords: uniform distribution of sequences
reviewed by: Tibor Šalát
MaRDI profile type: Publication
full work available at URL: aima.1995.1069
OpenAlex ID: W2014871121
DOI: 10.1006/AIMA.1995.1069
Recommended article: Isomorphisms for convergence structures. II: Borel spaces and infinite measures
Recommended article: Continuous transforms of a Lebesgue measure.
Recommended article: Some remarks on countably determined measures and uniform distribution of sequences
Recommended article: Q4895995
Recommended article: Topological Spaces with Skorokhod Representation Property
Recommended article: SURJECTIVE ISOMETRIES ON REARRANGEMENT-INVARIANT SPACES
Recommended article: Q4013116
Recommended article: On the existence of an invariant measure
Recommended article: On images of Borel measures under Borel mappings
description: scientific article; zbMATH DE number 863625