Spaceability and algebrability of sets of nowhere integrable functions

TL;DR

This paper investigates the spaceability and algebrability of various classes of nowhere integrable functions, establishing new results on their algebraic and topological structures, with applications to different integrability concepts.

Contribution

It improves existing results by demonstrating spaceability and strong algebrability of specific sets of nowhere integrable functions, and explores their structural properties in various function spaces.

Findings

Set of Lebesgue integrable functions with nowhere essentially bounded is spaceable.

Set of functions with dense jump discontinuities is strongly -algebrable.

Set of Kurzweil integrable but not Lebesgue integrable functions is spaceable but not 1-algebrable.

Abstract

We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269], and that it is strongly $\mathfrak{c}$-algebrable. We prove strong $\mathfrak{c}$-algebrability and non-separable 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 [V. I. Gurariy, \textit{Subspaces and bases in spaces of continuous functions (Russian),} Dokl. Akad. Nauk SSSR, \textbf{167} (1966), 971-973].