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), 20252037
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 which are nowhere essentially bounded is spaceable, improving a result from GarcíaPacheco, Martín, and SeoaneSepúlveda, 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 LebesguenowhereRiemann integrable and RiemannnowhereNewton 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
(2006i:26004), http://dx.doi.org/10.1090/S0002993904075331
 2.
Richard
M. Aron and Juan
B. SeoaneSepúlveda, Algebrability of the set of everywhere
surjective functions on ℂ, Bull. Belg. Math. Soc. Simon Stevin
14 (2007), no. 1, 25–31. MR 2327324
(2008d:26016)
 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
(2001g:26005), http://dx.doi.org/10.1112/S0024610700008905
 5.
P. Enflo and V. I. Gurariy, On lineability and spaceability of sets in function spaces, unpublished notes.
 6.
F.
J. GarcíaPacheco, M.
Martín, and J.
B. SeoaneSepúlveda, Lineability, spaceability, and
algebrability of certain subsets of function spaces, Taiwanese J.
Math. 13 (2009), no. 4, 1257–1269. MR 2543741
(2010h:46072)
 7.
F.
J. GarcíaPacheco, N.
Palmberg, and J.
B. SeoaneSepúlveda, Lineability and algebrability of
pathological phenomena in analysis, J. Math. Anal. Appl.
326 (2007), no. 2, 929–939. MR 2280953
(2007i:26003), http://dx.doi.org/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
(95m:26010)
 9.
V.
I. Gurariĭ, Subspaces and bases in spaces of continuous
functions, Dokl. Akad. Nauk SSSR 167 (1966),
971–973 (Russian). MR 0199674
(33 #7817)
 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
(92j:46046)
 1.
 R. Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on , Proc. Amer. Math. Soc., 133 (2005), 795803. MR 2113929 (2006i:26004)
 2.
 R. Aron and J. B. SeoaneSepúlveda, Algebrability of the set of everywhere surjective functions on , Bull. Belg. Math. Soc. Simon Stevin, 14 (2007), 2531. MR 2327324 (2008d:26016)
 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., 62 (2000), no. 1, 117126. MR 1771855 (2001g:26005)
 5.
 P. Enflo and V. I. Gurariy, On lineability and spaceability of sets in function spaces, unpublished notes.
 6.
 F. J. GarcíaPacheco, M. Martín, and J. B. SeoaneSepúlveda. Lineability, spaceability, and algebrability of certain subsets of function spaces, Taiwanese J. Math., 13 (2009), no. 4, 12571269. MR 2543741 (2010h:46072)
 7.
 F. J. GarcíaPacheco, N. Palmberg, and J. B. SeoaneSepúlveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl., 326 (2007), 929939. MR 2280953 (2007i:26003)
 8.
 R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math. Soc., Providence, RI, 1994. MR 1288751 (95m:26010)
 9.
 V. I. Gurariy, Subspaces and bases in spaces of continuous functions (Russian), Dokl. Akad. Nauk SSSR, 167 (1966), 971973. MR 0199674 (33:7817)
 10.
 V. I. Gurariy, Linear spaces composed of nondifferentiable functions, C. R. Acad. Bulgare Sci., 44 (1991), 1316. MR 1127022 (92j:46046)
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Additional Information
Szymon Głab
Affiliation:
Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93005 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 05508900, São Paulo, Brazil
Email:
plkaufmann@gmail.com
Leonardo Pellegrini
Affiliation:
Instituto de matemática e estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508900, São Paulo, Brazil
Email:
leonardo@ime.usp.br
DOI:
http://dx.doi.org/10.1090/S000299392012115746
PII:
S 00029939(2012)115746
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 22562009.
Communicated by:
Thomas Schlumprecht
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
