Spaceability and algebrability of sets of nowhere integrable functions

Gła̧b, Szymon; Kaufmann, Pedro; Pellegrini, Leonardo

doi:10.1090/S0002-9939-2012-11574-6

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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