{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:09:34Z","timestamp":1783904974842,"version":"3.55.0"},"reference-count":7,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous article [5], we constructed a one-dimensional 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.<\/jats:p>","DOI":"10.2478\/forma-2021-0018","type":"journal-article","created":{"date-parts":[[2022,7,9]],"date-time":"2022-07-09T16:34:35Z","timestamp":1657384475000},"page":"185-199","source":"Crossref","is-referenced-by-count":6,"title":["Relationship between the Riemann and Lebesgue Integrals"],"prefix":"10.2478","volume":"29","author":[{"given":"Noboru","family":"Endou","sequence":"first","affiliation":[{"name":"National Institute of Technology, Gifu College , 2236-2 Kamimakuwa, Motosu , Gifu , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2022,7,9]]},"reference":[{"key":"2026071215073907931_j_forma-2021-0018_ref_001","unstructured":"[1] Tom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969."},{"key":"2026071215073907931_j_forma-2021-0018_ref_002","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, 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\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071215073907931_j_forma-2021-0018_ref_003","doi-asserted-by":"crossref","unstructured":"[3] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071215073907931_j_forma-2021-0018_ref_004","doi-asserted-by":"crossref","unstructured":"[4] Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69\u201379, 2016. doi:10.1515\/forma-2016-0006.","DOI":"10.1515\/forma-2016-0006"},{"key":"2026071215073907931_j_forma-2021-0018_ref_005","doi-asserted-by":"crossref","unstructured":"[5] Noboru Endou. Reconstruction of the one-dimensional Lebesgue measure. Formalized Mathematics, 28(1):93\u2013104, 2020. doi:10.2478\/forma-2020-0008.","DOI":"10.2478\/forma-2020-0008"},{"key":"2026071215073907931_j_forma-2021-0018_ref_006","unstructured":"[6] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999."},{"key":"2026071215073907931_j_forma-2021-0018_ref_007","doi-asserted-by":"crossref","unstructured":"[7] 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\u2013236, 2007. doi:10.2478\/v10037-007-0026-3.","DOI":"10.2478\/v10037-007-0026-3"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0018","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:31:51Z","timestamp":1783902711000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0018"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,1]]},"references-count":7,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,7,9]]},"published-print":{"date-parts":[[2021,12,1]]}},"alternative-id":["10.2478\/forma-2021-0018"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0018","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,1]]}}}