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

Abstract

To demonstrate more visibly the close relation between the<br />continuity and integrability, a new proof for the Banach-Zarecki<br />theorem is presented on the basis of the Radon-Nikodym theorem<br />which emphasizes on measure-type properties of the Lebesgue<br />integral. The Banach-Zarecki theorem says that a real-valued<br />function $F$ is absolutely continuous on a finite closed interval<br />if and only if it is continuous and of bounded variation when it<br />satisfies Lusin's condition. In the present proof indeed a more<br />general result is obtained for the Jordan decomposition of $F$.