A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral

Abstract

Several new multidimensional integration theories that generalize or are analogous to the Denjoy-Perron-Henstock-Kurzweil integral were developed in recent years by several authors (e.g. [H], [L] (Generalized Riemann integral), [CD] (Generalized Denjoy integral), [M1] (GP-integral), [KMP] (BV-integral)). The purpose in the development of these theories has been to examine more general versions of the classical theorems in Lebesgue integration theory, such as the Divergence theorem, Fubini’s theorem, or convergence theorems. In this paper, we propose a simple elementary multidimensional integration that has a number of advantages also. First, as remarked in [L], unlike the onedimensional case, a drawback of the known multidimensional integrals is that one cannot develop in the same system both Divergence and Fubini type theorems. This can be done with the integral presented here. Second, one main goal of the above theories has been to weaken the smoothness condition on the vector fields in the Divergence theorem. The continuous differentiability of the vector fields was replaced by their continuity and their pointwise, or asymptotic, or a.e. differentiability (with some other supplementary conditions). Here, we can remove all hypotheses about differentiability and prove a Divergence theorem for the class of all continuous vector fields (in fact, for a larger class of distributions). The third point concerns convergence theorems. In some of the previous integration theories, the convergence theorems are rather complicated (see e.g. Ch. 5, [L]), and in some others, they seem