{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:51:29Z","timestamp":1777423889671,"version":"3.51.4"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2017,10,1]],"date-time":"2017-10-01T00:00:00Z","timestamp":1506816000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In this article, we formalize in the Mizar system [1, 7] the Lebesgue type integral and convergence theorems for non positive functions [8],[2]. Many theorems are based on our previous results [5], [6].<\/jats:p>","DOI":"10.1515\/forma-2017-0022","type":"journal-article","created":{"date-parts":[[2017,12,20]],"date-time":"2017-12-20T22:16:12Z","timestamp":1513808172000},"page":"227-240","source":"Crossref","is-referenced-by-count":1,"title":["Integral of Non Positive Functions"],"prefix":"10.1515","volume":"25","author":[{"given":"Noboru","family":"Endou","sequence":"first","affiliation":[{"name":"National Institute of Technology , Gifu College , 2236-2 Kamimakuwa, Motosu, Gifu , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,19]]},"reference":[{"key":"2021040800062260881_j_forma-2017-0022_ref_001_w2aab3b7b6b1b6b1ab1ab1Aa","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"},{"key":"2021040800062260881_j_forma-2017-0022_ref_002_w2aab3b7b6b1b6b1ab1ab2Aa","unstructured":"[2] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007."},{"key":"2021040800062260881_j_forma-2017-0022_ref_003_w2aab3b7b6b1b6b1ab1ab3Aa","unstructured":"[3] Noboru Endou. Extended real-valued double sequence and its convergence. Formalized Mathematics, 23(3):253\u2013277, 2015. doi:10.1515\/forma-2015-0021.10.1515\/forma-2015-0021"},{"key":"2021040800062260881_j_forma-2017-0022_ref_004_w2aab3b7b6b1b6b1ab1ab4Aa","unstructured":"[4] Noboru Endou. Fubini\u2019s theorem on measure. Formalized Mathematics, 25(1):1\u201329, 2017. doi:10.1515\/forma-2017-0001.10.1515\/forma-2017-0001"},{"key":"2021040800062260881_j_forma-2017-0022_ref_005_w2aab3b7b6b1b6b1ab1ab5Aa","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"},{"key":"2021040800062260881_j_forma-2017-0022_ref_006_w2aab3b7b6b1b6b1ab1ab6Aa","unstructured":"[6] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167\u2013175, 2008. doi:10.2478\/v10037-008-0023-1.10.2478\/v10037-008-0023-1"},{"key":"2021040800062260881_j_forma-2017-0022_ref_007_w2aab3b7b6b1b6b1ab1ab7Aa","unstructured":"[7] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1"},{"key":"2021040800062260881_j_forma-2017-0022_ref_008_w2aab3b7b6b1b6b1ab1ab8Aa","unstructured":"[8] P. R. Halmos. Measure Theory. Springer-Verlag, 1974."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/25\/3\/article-p227.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0022","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T05:30:15Z","timestamp":1617859815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,1]]},"references-count":8,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,12,19]]},"published-print":{"date-parts":[[2017,10,1]]}},"alternative-id":["10.1515\/forma-2017-0022"],"URL":"https:\/\/doi.org\/10.1515\/forma-2017-0022","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2017,10,1]]}}}