Relationship between the Riemann and Lebesgue Integrals

Pole DC	Wartość	Język
dc.contributor.author	Endou, Noboru	-
dc.date.accessioned	2022-07-22T10:33:13Z	-
dc.date.available	2022-07-22T10:33:13Z	-
dc.date.issued	2021	-
dc.identifier.citation	Formalized Mathematics, Volume 29, Issue 4, Pages 185-199	pl
dc.identifier.issn	1426-2630	-
dc.identifier.uri	http://hdl.handle.net/11320/13659	-
dc.description.abstract	The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous article [5], we constructed a onedimensional Lebesgue measure. The one-dimensional Lebesgue measure provides a measure of any intervals, which can be used to prove the well-known relationship [6] between the Riemann and Lebesgue integrals [1]. We also proved the relationship between the integral of a given measure and that of its complete measure. As the result of this work, the Lebesgue integral of a bounded real valued function in the Mizar system [2], [3] can be calculated by the Riemann integral.	pl
dc.language.iso	en	pl
dc.publisher	DeGruyter Open	pl
dc.rights	Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)	pl
dc.rights.uri	https://creativecommons.org/licenses/by-sa/3.0/	pl
dc.subject	Riemann integrals	pl
dc.subject	Lebesgue integrals	pl
dc.title	Relationship between the Riemann and Lebesgue Integrals	pl
dc.type	Article	pl
dc.rights.holder	© 2021 University of Białymstoku	pl
dc.rights.holder	CC-BY-SA License ver. 3.0 or later	pl
dc.identifier.doi	10.2478/forma-2021-0018	-
dc.description.Affiliation	National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, Japan	pl
dc.description.references	Tom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969.	pl
dc.description.references	Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.	pl
dc.description.references	Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.	pl
dc.description.references	Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69–79, 2016. doi:10.1515/forma-2016-0006.	pl
dc.description.references	Noboru Endou. Reconstruction of the one-dimensional Lebesgue measure. Formalized Mathematics, 28(1):93–104, 2020. doi:10.2478/forma-2020-0008.	pl
dc.description.references	Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999.	pl
dc.description.references	Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231–236, 2007. doi:10.2478/v10037-007-0026-3.	pl
dc.identifier.eissn	1898-9934	-
dc.description.volume	29	pl
dc.description.issue	4	pl
dc.description.firstpage	185	pl
dc.description.lastpage	199	pl
dc.identifier.citation2	Formalized Mathematics	pl
dc.identifier.orcid	0000-0002-5922-2332	-
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