A definition of integrals
In this paper a very general approach to Riemann integration is developed. The general idea is to associate with a real valued function Riemann sums depending on partitions that avoid small sets that contain the pathologies of the function. In order to do this the author needs families of partitions that are equipped with a filter, and a system of small sets that is a filter on the set of subsets of the space on which the functions are defined. The resulting definition of an integration space, a 6-tuple, and the associated integral are fairly abstract. Restrictions on the general theory give an absolute integral that is the Lebesgue integral with respect to a natural measure associated with the integration space. Another section considers the system of small sets defined by a gauge; and the resulting integral is the same as the associated gauge integral. In particular, an integral is obtained that is at least as general as the Kurzweil-Henstock integral. The concepts and definitions are too many and detailed to be discussed here but all those working in the theory of general integrals should read this very interesting paper. The full connections with the Henstock theory remain to be developed.
scientific article; zbMATH DE number 861800