Anti-Fubini and pseudo-Fubini functions
type: Item
instance of: scholarly article
title: Anti-Fubini and pseudo-Fubini functions
zbMATH Open document ID: 1482.46023
author: L. Bernal-González
author: María del Carmen Calderón-Moreno
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2021-06-30T00:00:00Z
review text: Fubini's theorem asserts that if a measurable function of two variables is integrable with respect to the product measure, then the iterated integrals coincide. In other words, if the iterated integrals do not coincide, then the function is not integrable for the product measure. The converse of Fubini's theorem is not true, in general. So, there exist measurable functions of two variables which are not integrable but the iterated integrals exist as real numbers and share the same value. In this paper the authors investigate the topological and algebraic size of the following classes of functions: \begin{itemize} \item[(1)] Measurable, non-integrable functions of two variables such that their iterated integrals do not coincide. \item[(2)] Measurable, non-integrable functions of two variables whose iterated integrals exist as real numbers and share the same value. \end{itemize} The topological size is investigated by means of the notion of residuality and the algebraic size is explored using the concepts of lineability and spaceability. The measures considered in the paper are \(\sigma\)-finite and the class of functions satisfying (1) is denoted by \(\alpha\mathcal{F}\) (anti-Fubini) and the class of functions satisfying (2) is represented by \(p\mathcal{F}\) (pseudo-Fubini). Several results on the size of \(\alpha\mathcal{F}\) and \(p\mathcal{F}\) are presented in the paper. For instance, it is shown that \(\alpha\mathcal{F}\) is not \(2\)-lineable, which is specially interesting because this kind of result is not common in the literature.
Mathematics Subject Classification ID: 46B87
Mathematics Subject Classification ID: 28A35
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7365225
zbMATH Keywords: product measure
zbMATH Keywords: Fubini's theorem
zbMATH Keywords: lineability
zbMATH Keywords: residuality
reviewed by: Daniel Pellegrino
MaRDI profile type: Publication
full work available at URL: s13398-021-01067-7
OpenAlex ID: W3166318962
cites work: Large structures in certain subsets of Orlicz spaces
cites work: Lineability
cites work: On dense-lineability of sets of functions on \(\mathbb R\)
cites work: Lineability within Peano curves, martingales, and integral theory
cites work: Algebraic genericity of strict-order integrability
cites work: On the dimension of subspaces of continuous functions attaining their maximum finitely many times
cites work: Large algebras of singular functions vanishing on prescribed sets
cites work: Spaceability of strict order integrability
cites work: Lineability criteria, with applications
cites work: Maximal spaceability in sequence spaces
cites work: Distinguished subspaces of Lpof maximal dimension
cites work: On very non-linear subsets of continuous functions
cites work: Measure Theory
cites work: Q4255465
cites work: Large structures made of nowhere Lqfunctions
cites work: Spaceability and algebrability of sets of nowhere integrable functions
cites work: Q5577487
cites work: On lineability of sets of continuous functions
cites work: Q3219291
cites work: Q3705984
cites work: Operator ranges and spaceability
cites work: Set theory. An introduction to independence proofs
cites work: Lineability in subsets of measure and function spaces
cites work: Q4339232
cites work: Separability of the L1-space of a vector measure
cites work: Q4257161
cites work: Nonlinear subsets of function spaces and spaceability
cites work: Q5563474
DOI: 10.1007/S13398-021-01067-7
description: scientific article; zbMATH DE number 7365225