On additive absolutely nonmeasurable Sierpiński-Zygmund functions
type: Item
instance of: scholarly article
title: On additive absolutely nonmeasurable Sierpiński-Zygmund functions
zbMATH Open document ID: 1114.28001
published in: Real Analysis Exchange
publication date: 2006-11-13T00:00:00Z
review text: Let \(E\) be a nonempty set, \(M\) be a set of real measures defined on \(E\) (measures defined on \(\sigma\)-algebras of subsets of \(E\)) and \(f: E\to\mathbb{R}\) be a function. The function \(f\) is called absolutely nonmeasurable with respect to \(M\) if \(f\) is nonmeasurable with respect to any measure from \(M\). Denote by \(M(E)\) the set of nonzero \(\sigma\)-finite diffused measures on \(E\) (i.e. vanishing on all singletons). The function \(f\) is called absolutely nonmeasurable if \(f\) is absolutely nonmeasurable with respect to \(M(E)\). A set \(X\) in \(\mathbb{R}\) is called a universal measure zero set if \(\mu^*(X)= 0\) for every \(\sigma\)-finite diffused Borel measure \(\mu\) on \(\mathbb{R},\mu^*\) being the outer measure associated with \(\mu\). The author uses the result that \(f\) is absolutely nonmeasurable if and only if the range of \(f\) is a universal measure zero subset of \(\mathbb{R}\) and \((\forall t\in\mathbb{R}) (\text{card}\{f^{-1}(t)\}\leq\omega),\) where \(\omega\) is a countable cardinal. A function \(f: \mathbb{R}\to\mathbb{R}\) is called a Vitali type function if \((\forall x\in \mathbb{R})(f(x)- x\in Q)\) and \(\text{ran}(f)\) is a selector of the partition \(\mathbb{R}/Q\). Then, if \(f\) is a Vitali type function, the Lebesgue measure \(\lambda\) can be extended to a measure \(\mu\) on \(\mathbb{R}\) such that \(f\) turns out to be measurable with respect to \(\mu\) (therefore, \(f\) is not absolutely nonmeasurable with respect to the class of all extensions of \(\lambda\)). It is known that \(f: \mathbb{R}\to\mathbb{R}\) is a solution of the Cauchy functional equation \((\forall(x,y)\in \mathbb{R}^2)(f(x+ y)= f(x)+ f(y))\) if and only if \(f\) is a homomorphism of the additive group \(\mathbb{R}\) into itself. A function \(f: \mathbb{R}\to\mathbb{R}\) is called a Sierpiński-Zygmund function if \((\forall Y\subset\mathbb{R})(\text{card\,}Y= c)(f/Y\text{ is not continuous on }Y),\) where \(c\) is the cardinality of the co…
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 5071514
zbMATH Keywords: absolutely nonmeasurable function
zbMATH Keywords: additive function
zbMATH Keywords: universal measure zero set
zbMATH Keywords: extension of measure
author: Alexander Kharazishvili
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
full work available at URL: realanalexch.31.2.0553
OpenAlex ID: W1554373578
DOI: 10.14321/REALANALEXCH.31.2.0553
description: scientific article; zbMATH DE number 5071514