A new proof for the Banach-Zarecki theorem: A light on integrability and continuity

TL;DR

This paper presents a new proof of the Banach-Zarecki theorem, highlighting the connection between continuity and integrability through measure-theoretic properties and the Radon-Nikodym theorem.

Contribution

It offers a novel proof based on measure theory, emphasizing the measure-type properties of the Lebesgue integral to clarify the relation between continuity and integrability.

Findings

New proof of Banach-Zarecki theorem using Radon-Nikodym theorem

Highlights measure-type properties of Lebesgue integral

Clarifies the relation between continuity and integrability

Abstract

To demonstrate more visibly the close relation between the continuity and integrability, a new proof for the Banach-Zarecki theorem is presented on the basis of the Radon-Nikodym theorem which emphasizes on measure-type properties of the Lebesgue integral. The Banach-Zarecki theorem says that a real-valued function F is absolutely continuous on a finite closed interval if and only if it is continuous and of bounded variation when it satisfies Lusin's condition (N).