Henstock-Kurzweil Integration on Metric Spaces Revisited

Abstract

We fill up some gaps in the existing literature on the Henstock-Kurzweil integration on metric measure spaces. The most important one is the choice of suitable candidates for `intervals' in metric spaces for which the conclusion of Cousin's lemma holds. We also provide some characterizations of compactness and completeness in terms of Cousin's lemma, along with some alternative proofs of a few related results. Then we improve the result regarding differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces, and a few consequences. We propose shorter proofs of measure theoretic characterizations of the HK-integral in terms of the variational measure $V_F$ as well as the $ACG^\Delta$ functions. Finally, we present alternative proofs of some generalizations of two results on Lebesgue integration. The first one bypasses Vitali covering lemma for a result on absolute continuity, while the second one is a version of the fundamental theorem of calculus in our setting.