Monotonically controlled integrals

Abstract

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

Final version will appear in the Volume commemorating Solomon Marcus