Non-absolute integration and application to Young geometric integration

TL;DR

This paper surveys non-absolutely convergent integrals like Henstock-Kurzweil and Pfeffer, and applies these ideas to develop multidimensional Young geometric integration using chains and cochains.

Contribution

It introduces a framework for multidimensional Young geometric integration based on non-absolute integrals and geometric chain theory, extending classical concepts.

Findings

Developed a new approach to multidimensional Young integration

Extended geometric integration theory to higher dimensions

Provided foundational results on generalized differential forms

Abstract

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on Young geometric integration, namely the integration of certain generalized differential forms over $m$-dimensional subsets of $\mathbb{R}^d$. This is achieved by introducing appropriate notions of chains and cochains, in the spirit of Whitney's geometric integration theory.