Monotonically Controlled Integrals

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