{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:10:37Z","timestamp":1783901437139,"version":"3.55.0"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"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":[[2019,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini\u2019s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini\u2019s theorem (30) was proved in complete form by this article.<\/jats:p>","DOI":"10.2478\/forma-2019-0007","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T05:34:02Z","timestamp":1558071242000},"page":"67-74","source":"Crossref","is-referenced-by-count":4,"title":["Fubini\u2019s Theorem"],"prefix":"10.2478","volume":"27","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":[[2019,5,16]]},"reference":[{"key":"2026071214315408471_j_forma-2019-0007_ref_001_w2aab3b7b6b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. 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":"2026071214315408471_j_forma-2019-0007_ref_002_w2aab3b7b6b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.10.1515\/9783110866209","DOI":"10.1515\/9783110866209"},{"key":"2026071214315408471_j_forma-2019-0007_ref_003_w2aab3b7b6b1b6b1ab1ab3Aa","unstructured":"[3] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007."},{"key":"2026071214315408471_j_forma-2019-0007_ref_004_w2aab3b7b6b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","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","DOI":"10.1515\/forma-2017-0001"},{"key":"2026071214315408471_j_forma-2019-0007_ref_005_w2aab3b7b6b1b6b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] Noboru Endou. Integral of non positive functions. Formalized Mathematics, 25(3):227\u2013240, 2017. doi:10.1515\/forma-2017-0022.10.1515\/forma-2017-0022","DOI":"10.1515\/forma-2017-0022"},{"key":"2026071214315408471_j_forma-2019-0007_ref_006_w2aab3b7b6b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] Noboru Endou. Fubini\u2019s theorem for non-negative or non-positive functions. Formalized Mathematics, 26(1):49\u201367, 2018. doi:10.2478\/forma-2018-0005.10.2478\/forma-2018-0005","DOI":"10.2478\/forma-2018-0005"},{"key":"2026071214315408471_j_forma-2019-0007_ref_007_w2aab3b7b6b1b6b1ab1ab7Aa","doi-asserted-by":"crossref","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","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026071214315408471_j_forma-2019-0007_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\/27\/1\/article-p67.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:55:31Z","timestamp":1783900531000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,1]]},"references-count":8,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,5,16]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.2478\/forma-2019-0007"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0007","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,4,1]]}}}