{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:11:09Z","timestamp":1783897869063,"version":"3.55.0"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In the Mizar system ([1], [2]), J\u00f3zef Bia\u0142as has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Bia\u0142as limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].<\/jats:p>","DOI":"10.2478\/forma-2020-0008","type":"journal-article","created":{"date-parts":[[2020,6,2]],"date-time":"2020-06-02T04:52:28Z","timestamp":1591073548000},"page":"93-104","source":"Crossref","is-referenced-by-count":6,"title":["Reconstruction of the One-Dimensional Lebesgue Measure"],"prefix":"10.2478","volume":"28","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":[[2020,5,29]]},"reference":[{"key":"2026071215073328365_j_forma-2020-0008_ref_001_w2aab3b7c15b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] 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_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071215073328365_j_forma-2020-0008_ref_002_w2aab3b7c15b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] 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\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071215073328365_j_forma-2020-0008_ref_003_w2aab3b7c15b1b6b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.10.1515\/9783110866209","DOI":"10.1515\/9783110866209"},{"key":"2026071215073328365_j_forma-2020-0008_ref_004_w2aab3b7c15b1b6b1ab1ab4Aa","unstructured":"[4] J\u00f3zef Bia\u0142as. The one-dimensional Lebesgue measure. Formalized Mathematics, 5(2):253\u2013258, 1996."},{"key":"2026071215073328365_j_forma-2020-0008_ref_005_w2aab3b7c15b1b6b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53\u201370, 2006. doi:10.2478\/v10037-006-0008-x.10.2478\/v10037-006-0008-x","DOI":"10.2478\/v10037-006-0008-x"},{"key":"2026071215073328365_j_forma-2020-0008_ref_006_w2aab3b7c15b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Hopf extension theorem of measure. Formalized Mathematics, 17(2):157\u2013162, 2009. doi:10.2478\/v10037-009-0018-6.10.2478\/v10037-009-0018-6","DOI":"10.2478\/v10037-009-0018-6"},{"key":"2026071215073328365_j_forma-2020-0008_ref_007_w2aab3b7c15b1b6b1ab1ab7Aa","unstructured":"[7] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999."},{"key":"2026071215073328365_j_forma-2020-0008_ref_008_w2aab3b7c15b1b6b1ab1ab8Aa","doi-asserted-by":"crossref","unstructured":"[8] 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\/v10037-007-0026-3.10.2478\/v10037-007-0026-3","DOI":"10.2478\/v10037-007-0026-3"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/content.sciendo.com\/view\/journals\/forma\/28\/1\/article-p93.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2020-0008","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:03:53Z","timestamp":1783897433000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2020-0008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,1]]},"references-count":8,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,5,29]]},"published-print":{"date-parts":[[2020,4,1]]}},"alternative-id":["10.2478\/forma-2020-0008"],"URL":"https:\/\/doi.org\/10.2478\/forma-2020-0008","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2020,4,1]]}}}