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Assistant Professor, Iran University of Science and Technology
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. In the present proof indeed a more general result is obtained for the Jordan decomposition of $F$.
Mahdipour Shirayeh, A. & Eshraghi, H. (2013). A new proof for the Banach-Zarecki theorem: A light
on integrability and continuity. Bulletin of the Iranian Mathematical Society, 39(5), 805–819.
MLA
Mahdipour Shirayeh, A., & Eshraghi, H. "A new proof for the Banach-Zarecki theorem: A light
on integrability and continuity", Bulletin of the Iranian Mathematical Society, 39, 5, 2013, 805–819.
HARVARD
Mahdipour Shirayeh, A., & Eshraghi, H. (2013). 'A new proof for the Banach-Zarecki theorem: A light
on integrability and continuity', Bulletin of the Iranian Mathematical Society, 39(5), pp. 805–819.
CHICAGO
Mahdipour Shirayeh, A. & Eshraghi, H., "A new proof for the Banach-Zarecki theorem: A light
on integrability and continuity." Bulletin of the Iranian Mathematical Society, 39 5 (2013): 805–819,
VANCOUVER
Mahdipour Shirayeh A., Eshraghi H. A new proof for the Banach-Zarecki theorem: A light
on integrability and continuity. BIMS. 2013;39(5):805–819.
