Algebraic genericity within the class of sup-measurable functions
type: Item
instance of: scholarly article
title: Algebraic genericity within the class of sup-measurable functions
zbMATH Open document ID: 1429.28001
author: L. Bernal-González
author: D. L. Rodríguez-Vidanes
author: G. A. Muñoz-Fernández
author: J. B. Seoane-Sepúlveda
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: A function \(F:\mathbb{R}^2\to\mathbb{R}\) is Lebesgue (Baire) sup-measurable if, for each Lebesgue (Baire) measurable function \(f:\mathbb{R}\to\mathbb{R}\), the superposition \(F_f:\mathbb{R}\to\mathbb{R}\), \(F_f(x)=F(x,f(x))\), is Lebesgue (Baire) measurable. The problem of existence of non-measurable sup-measurable functions is undecidable in ZFC. Under assumption of CH (or some weaker set-theoretical assumptions) non-measurable sup-measurable functions were constructed: \(\bullet\) in the measure case by \textit{Z. Grande} and \textit{J. S. Lipinski} [Colloq. Math. 39, 77--79 (1978; Zbl 0386.28006)], and independently, by \textit{A. B. Kharazishvili} [Soobshch. Akad. Nauk Gruz. SSR 100, 533--536 (1980; Zbl 0454.28005)], see also [\textit{A. B. Kharazishvili}, J. Appl. Anal. 3, No. 2, 211--223 (1997; Zbl 0901.28002)]; \(\bullet\) in the category case by \textit{E. Grande} and \textit{Z. Grande} [Fundam. Math. 121, 199--211 (1984; Zbl 0573.28007)]. On the other hand, the model of ZFC without of non-measurable sup-measurale functions was constructed \(\bullet\) in the measure case by \textit{A. Rosłanowski} and \textit{S. Shelah} [Isr. J. Math. 151, 61--110 (2006; Zbl 1125.03036)]; \(\bullet\) in the category case by \textit{K. Ciesielski} and \textit{S. Shelah} [J. Appl. Anal. 6, No. 2, 159--172 (2000; Zbl 0972.03048)]. In this paper the authors show that under CH (or some weaker set-theoretical assumptions) there is a continuum-dimensional vector space contained in the family of all Lebesgue (Baire) non-measurable sup-measurable functions (plus the zero function). In the second part of the paper, the authors consider also the families of weakly sup-measurable functions with respect to measure or category. A function \(F:\mathbb{R}^2\to\mathbb{R}\) is weakly sup-measurable if, for every continuous, differentiable almost everywhere function \(f:\mathbb{R}\to\mathbb{R}\), the superposition \(F_f\) is measurable, see [\textit{A. B. Kharazishvili}, J. Appl. Anal. 3,…
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 28A10
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 03E75
zbMATH DE Number: 7138436
zbMATH Keywords: lineability
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Baire property
zbMATH Keywords: category and measure
zbMATH Keywords: sup-measurable functions
zbMATH Keywords: non-measurable functions
zbMATH Keywords: Caratheodory superposition
MaRDI profile type: Publication
full work available at URL: j.jmaa.2019.123576
OpenAlex ID: W2979839564
cites work: Lineability
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Riemann integrability and Lebesgue measurability of the composite function
cites work: On the sup-measurable functions problem
cites work: Independent Bernstein sets and algebraic constructions
cites work: Strong algebrability of sets of sequences and functions
cites work: Algebrability, non-linear properties, and special functions
cites work: Highly tempering infinite matrices
cites work: \([\mathcal{S}]\)-linear and convex structures in function families
cites work: Lineability criteria, with applications
cites work: Linear subsets of nonlinear sets in topological vector spaces
cites work: Basic sequences and spaceability in \(\ell_p\) spaces
cites work: Algebras of measurable extendable functions of maximal cardinality
cites work: Differentiability versus continuity: Restriction and extension theorems and monstrous examples
cites work: Some results and open questions on spaceability in function spaces
cites work: On the zero-set of 2-homogeneous polynomials in Banach spaces
cites work: Sierpiński-Zygmund functions and other problems on lineability
cites work: Lineability and additivity in \(\mathbb R^{\mathbb R}\)
cites work: An undecidable case of lineability in \(\mathbb R^{\mathbb R}\)
cites work: Vector spaces of non-measurable functions
cites work: Q5577487
cites work: Q4031612
cites work: On lineability of sets of continuous functions
cites work: Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions
cites work: Q4759444
cites work: Q2959041
cites work: Q3715106
cites work: Corrections and Additions to "Some Properties of Measure and Category"
cites work: Some Properties of Measure and Category
cites work: Q3874454
cites work: Bounded linear non-absolutely summing operators
cites work: Measured creatures
Wikidata QID: Q127130419
DOI: 10.1016/J.JMAA.2019.123576
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description: scientific article; zbMATH DE number 7138436