Vector spaces of non-measurable functions
type: Item
instance of: scholarly article
title: Vector spaces of non-measurable functions
zbMATH Open document ID: 1126.28002
publication date: 2007-05-24T00:00:00Z
review text: Let \((\Omega,\Sigma)\) be a measurable space and let \(\gamma\) be an infinite cardinal. Suppose there is a family \({\mathcal V}\) of pairwise disjoint subsets of \(\Omega\) with card \({\mathcal V}=\gamma\) such that the union of each nonempty subfamily of \({\mathcal V}\) is not in \(\Sigma\). Then for every Banach space \(X\) of density character \(\gamma\) there exists an injective continuous linear operator \(T: X\to\mathbb{K}^\Omega\), where \(\mathbb{K}\) is the scalar field of \(X\), with \(T(x)\) not \(\Sigma\)-measurable for each \(x\neq 0\) (Proposition 2.4). An isometric version of this result is also established. The paper is written carelessly. In particular, the crucial disjointness condition is missing in the formulations of Lemma 2.1, Theorem 2.2 and Propositions 2.3 and 2.4. On the other hand, the non-measurability assumption in Proposition 2.3 is irrelevant. Also, in the course of the proof of Proposition 2.4 the authors establish a result which is an obvious consequence of a standard proposition formulated at the end of the proof of Theorem 2.5. Finally, we note that the result established after the proof of Lemma 2.1 is well known. For example, according to a classical theorem due to \textit{N. Lusin} and \textit{W. Sierpiński} [C.R. 165, 422-424 (1917; JFM 46.0294.01)] the unit interval can be partitioned into \({\mathfrak c}\)-many sets of Lebesgue outer measure~1.
reviewed by: Zbigniew Lipecki
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46E99
zbMATH DE Number: 5156877
zbMATH Keywords: non-measurable function
zbMATH Keywords: Banach space
zbMATH Keywords: density character
zbMATH Keywords: lineability
zbMATH Keywords: spaceability
author: Francisco García-Pacheco
author: J. B. Seoane-Sepúlveda
MaRDI profile type: Publication
full work available at URL: s10114-005-0726-y
OpenAlex ID: W2072463698
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Every Separable Banach Space is Isometric to a Space of Continuous Nowhere Differentiable Functions
cites work: Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions
cites work: Algebrability of the set of everywhere surjective functions on \(\mathbb C\)
cites work: Q4783634
cites work: Classical Banach spaces. 1: Sequence spaces. 2. Function spaces.
DOI: 10.1007/S10114-005-0726-Y
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5156877