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

Document Type : Research Paper


1 Postdoctoral Researcher, Brock University, Canada

2 Assistant Professor, Iran University of Science and Technology


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. In the present proof indeed a more
general result is obtained for the Jordan decomposition of $F$.


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