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

MathSciNet review:
3034428

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the set of Lebesgue integrable functions in which are nowhere essentially bounded is spaceable, improving a result

from García-Pacheco, Martín, and Seoane-Sepúlveda, and that it is strongly

-algebrable. We prove strong -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 -algebrable. We also show that there exists an infinite dimensional vector space of differentiable functions such that each element of the -closure of is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from Gurariy.

**1.**Richard Aron, V. I. Gurariy, and J. B. Seoane,*Lineability and spaceability of sets of functions on ℝ*, Proc. Amer. Math. Soc.**133**(2005), no. 3, 795–803 (electronic). MR**2113929**, 10.1090/S0002-9939-04-07533-1**2.**Richard M. Aron and Juan B. Seoane-Sepúlveda,*Algebrability of the set of everywhere surjective functions on ℂ*, Bull. Belg. Math. Soc. Simon Stevin**14**(2007), no. 1, 25–31. MR**2327324****3.**A. Bartoszewicz and Sz. Głab,*Strong algebrability of sets of sequences and functions,*Proc. Amer. Math. Soc. (to appear).**4.**B. Bongiorno, L. Di Piazza, and D. Preiss,*A constructive minimal integral which includes Lebesgue integrable functions and derivatives*, J. London Math. Soc. (2)**62**(2000), no. 1, 117–126. MR**1771855**, 10.1112/S0024610700008905**5.**P. Enflo and V. I. Gurariy,*On lineability and spaceability of sets in function spaces,*unpublished notes.**6.**F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda,*Lineability, spaceability, and algebrability of certain subsets of function spaces*, Taiwanese J. Math.**13**(2009), no. 4, 1257–1269. MR**2543741****7.**F. J. García-Pacheco, N. Palmberg, and J. B. Seoane-Sepúlveda,*Lineability and algebrability of pathological phenomena in analysis*, J. Math. Anal. Appl.**326**(2007), no. 2, 929–939. MR**2280953**, 10.1016/j.jmaa.2006.03.025**8.**Russell A. Gordon,*The integrals of Lebesgue, Denjoy, Perron, and Henstock*, Graduate Studies in Mathematics, vol. 4, American Mathematical Society, Providence, RI, 1994. MR**1288751****9.**V. I. Gurariĭ,*Subspaces and bases in spaces of continuous functions*, Dokl. Akad. Nauk SSSR**167**(1966), 971–973 (Russian). MR**0199674****10.**V. I. Gurariĭ,*Linear spaces composed of everywhere nondifferentiable functions*, C. R. Acad. Bulgare Sci.**44**(1991), no. 5, 13–16 (Russian). MR**1127022**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
26A30,
26A42,
26A39,
26A45

Retrieve articles in all journals with MSC (2010): 26A30, 26A42, 26A39, 26A45

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

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.