The Kolmogorov Ideas on the Integration Theory in Modern Research

Generalizations of construction of Kolmogorov integral to the case of Banach space-valued functions are considered. We demonstrate how the Kolmogorov ideas on integration theory, in particular, the notion of differential equivalence, have been developed in the theory of the Henstock–Kurzweil integral. In this connection, a variational version of a Henstock type integral with respect to a rather general derivation basis is studied. An example of application of this integral to harmonic analysis is given. Some results related to the Kolmogorov -integral are also considered.