Negligible Variation and the Change of Variables Theorem

TL;DR

This paper establishes necessary and sufficient conditions for the change of variables formula in the Henstock-Kurzweil (HK) integral, impacting the understanding of the Fundamental Theorem of Calculus within this framework.

Contribution

It provides a new characterization of when the change of variables formula holds for the HK integral, linking it to the Lebesgue integral and the Fundamental Theorem of Calculus.

Findings

Derived necessary and sufficient conditions for change of variables in HK integral.

Connected conditions for HK integral to the Lebesgue integral.

Established criteria for the Fundamental Theorem of Calculus in HK context.

Abstract

In this note we prove a necessary and sufficient condition for the change of variables formula for the HK integral, with implications for the change of variables formula for the Lebesgue integral. As a corollary, we obtain a necessary and sufficient condition for the Fundamental Theorem of Calculus to hold for the HK integral.