Monotonically controlled integrals

TL;DR

This paper explores the family of alpha-monotonically controlled integrals, revealing how their relationship to classical integrals varies with the parameter alpha, including cases where they coincide or differ significantly.

Contribution

It introduces and analyzes the entire scale of alpha-monotonically controlled integrals, clarifying their dependence on alpha and their relation to classical integrals.

Findings

For alpha<1, they do not include the Lebesgue integral.

For 1<=alpha<=2, they coincide with the Denjoy-Perron integral.

For alpha>2, they are distinct from and not contained in the Denjoy-Khintchine integral.

Abstract

The monotonically controlled integral defined by Bendov\'a and Mal\'y, which is equivalent to the Denjoy-Perron integral, admits a natural parameter $\alpha>0$ thereby leading to the whole scale of integrals called $\alpha$-monotonically controlled integrals. While the power of these integrals is easily seen to increase with increasing $\alpha$, our main results show that their exact dependence on $\alpha$ is rather curious. For $\alpha<1$ they do not even contain the Lebesgue integral, for $1\le\alpha\le 2$ they coincide with the Denjoy-Perron integral, and for $\alpha>2$ they are mutually different and not even contained in the Denjoy-Khintchine integral.