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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spaceability and algebrability of sets of nowhere integrable functions


Authors: Szymon Głab, Pedro L. Kaufmann and Leonardo Pellegrini
Journal: Proc. Amer. Math. Soc. 141 (2013), 2025-2037
MSC (2010): Primary 26A30; Secondary 26A42, 26A39, 26A45
Published electronically: December 28, 2012
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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 Głab
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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11574-6
PII: S 0002-9939(2012)11574-6
Keywords: Spaceability, algebrability, nowhere integrable functions, bounded variation functions
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.