On lineability of families of non-measurable functions of two variable
type: Item
instance of: scholarly article
title: On lineability of families of non-measurable functions of two variable
zbMATH Open document ID: 1471.28002
author: Tomasz Natkaniec
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2021-05-31T00:00:00Z
review text: Given a set \(X\) and a class~\(C\) of real-valued functions on~\(X\) one says that \(C\)~is \(\kappa\)-lineable (where \(\kappa\) is some cardinal) if \(C\cup\{0\}\) contains a \(\kappa\)-dimensional vector subspace of~\(\mathbb{R}^X\). Many classes are lineable simply because they are vector spaces (continuous, integrable, differentiable); hence lineability becomes non-trivial\slash interesting for classes that are non automatically closed under linear combinations. The author considers the class of functions on~\(\mathbb{R}^2\) that are sup-measurable: \(F:\mathbb{R}^2\to\mathbb{R}\) is sup-measurable if \(x\mapsto F(x,f(x))\) is measurable for every measurable function~\(f\) (`sup' is short for `superposition'). The Continuum Hypothesis implies that the class of sup-measurable but not measurable functions is \(2^\mathfrak{c}\)-lineable. A similar result obtains in ZFC for functions that are not measurable but all of whose vertical and horizontal sections are measurable.\par The paper contains many more results of this nature.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E65
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 7352841
zbMATH Keywords: lineability
zbMATH Keywords: function of two variables
zbMATH Keywords: sup-measurable function
zbMATH Keywords: non-measurable function
zbMATH Keywords: Sierpiński set
zbMATH Keywords: separately measurable function
MaRDI profile type: Publication
full work available at URL: s13398-020-00980-7
OpenAlex ID: W3120243077
cites work: Lineability
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Some remarks on sup-measurability
cites work: On the sup-measurable functions problem
cites work: Independent Bernstein sets and algebraic constructions
cites work: Algebrability, non-linear properties, and special functions
cites work: Linear subsets of nonlinear sets in topological vector spaces
cites work: Algebraic genericity within the class of sup-measurable functions
cites work: Differentiation of real functions
cites work: Q4211175
cites work: On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity
cites work: Restricted continuity and a theorem of Luzin
cites work: Q5669018
cites work: Q5677739
cites work: Q4773925
cites work: Semiéquicontinuité approximative et measurabilité
cites work: Un exemple d'une fonction sup-mesurable qui n'est pas mesurable
cites work: Q4759444
cites work: Sur les opérations linéaires dans l'espace des fonctions bornées
cites work: Q3901732
cites work: Sup-Measurable and Weakly Sup-Measurable Mappings in the Theory of Ordinary Differential Equations
cites work: Q2959041
cites work: Q5541832
cites work: Measurability of functions with approximately continuous vertical sections and measurable horizontal sections
cites work: Q5635659
cites work: Q3715868
cites work: Measured creatures
cites work: Sur La Premiere Derivee
DOI: 10.1007/S13398-020-00980-7
description: scientific article; zbMATH DE number 7352841