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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Spaceability and algebrability of sets of nowhere integrable functions
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by Szymon Gła̧b, Pedro L. Kaufmann and Leonardo Pellegrini PDF
Proc. Amer. Math. Soc. 141 (2013), 2025-2037 Request permission

Abstract:

We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from García-Pacheco, Martín, and Seoane-Sepúlveda, and that it is strongly $\mathfrak {c}$-algebrable. We prove strong $\mathfrak {c}$-algebrability and nonseparable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesgue-nowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition, we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not $1$-algebrable. We also show that there exists an infinite dimensional vector space $S$ of differentiable functions such that each element of the $C([0,1])$-closure of $S$ is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from Gurariy.
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Additional Information
  • Szymon Gła̧b
  • Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
  • Email: szymon.glab@p.lodz.pl
  • Pedro L. Kaufmann
  • Affiliation: Instituto de matemática e estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-900, São Paulo, Brazil
  • MR Author ID: 998841
  • ORCID: 0000-0001-6175-7144
  • Email: plkaufmann@gmail.com
  • Leonardo Pellegrini
  • Affiliation: Instituto de matemática e estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-900, São Paulo, Brazil
  • Email: leonardo@ime.usp.br
  • Received by editor(s): September 23, 2011
  • Published electronically: December 28, 2012
  • Additional Notes: The second author was supported by CAPES, Research Grant PNPD 2256-2009.
  • Communicated by: Thomas Schlumprecht
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2025-2037
  • MSC (2010): Primary 26A30; Secondary 26A42, 26A39, 26A45
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11574-6
  • MathSciNet review: 3034428