Kolmogorov's ideas on the theory of integral 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 an application of this integral in harmonic analysis is given. Some results related to Kolmogorov $A$-integral are also considered.