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

2012/06/12 by Ali Mahdipour-Shirayeh, Mahdipour-Shirayeh, Ali, Homayoon Eshraghi +1
Mathematics · Physics and Astronomy · #26B30 #46G12 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:26B30 #msc:46G12

paper · pdf · doi:10.48550/arxiv.1206.2602

15 pages; Published in the Bulletin of Iranian Mathematical Society, 2012

arxiv created 2012/06/12 · arxiv updated 2012/06/13

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).

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