{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T02:08:20Z","timestamp":1783908500880,"version":"3.55.0"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2018,4,1]],"date-time":"2018-04-01T00:00:00Z","timestamp":1522540800000},"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":[[2018,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>The goal of this article is to show Fubini\u2019s theorem for non-negative or non-positive measurable functions [10], [2], [3], using the Mizar system [1], [9]. We formalized Fubini\u2019s theorem in our previous article [5], but in that case we showed the Fubini\u2019s theorem for measurable sets and it was not enough as the integral does not appear explicitly.<\/jats:p>\n                  <jats:p>\n                    On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space\n                    <jats:italic>\n                      L\n                      <jats:sup>p<\/jats:sup>\n                    <\/jats:italic>\n                    [12]. It should be mentioned also that H\u00f6lzl and Heller [11] have introduced the Lebesgue integration theory in Isabelle\/HOL and have proved Fubini\u2019s theorem there.\n                  <\/jats:p>","DOI":"10.2478\/forma-2018-0005","type":"journal-article","created":{"date-parts":[[2018,8,2]],"date-time":"2018-08-02T20:32:23Z","timestamp":1533241943000},"page":"49-67","source":"Crossref","is-referenced-by-count":1,"title":["Fubini\u2019s Theorem for Non-Negative or Non-Positive Functions"],"prefix":"10.2478","volume":"26","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":[[2018,7,28]]},"reference":[{"key":"2026071213090505446_j_forma-2018-0005_ref_001_w2aab3b7b4b1b6b1ab1ab1Aa","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":"2026071213090505446_j_forma-2018-0005_ref_002_w2aab3b7b4b1b6b1ab1ab2Aa","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":"2026071213090505446_j_forma-2018-0005_ref_003_w2aab3b7b4b1b6b1ab1ab3Aa","unstructured":"[3] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007."},{"key":"2026071213090505446_j_forma-2018-0005_ref_004_w2aab3b7b4b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69\u201379, 2016. doi:10.1515\/forma-2016-0006.10.1515\/forma-2016-0006","DOI":"10.1515\/forma-2016-0006"},{"key":"2026071213090505446_j_forma-2018-0005_ref_005_w2aab3b7b4b1b6b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] 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":"2026071213090505446_j_forma-2018-0005_ref_006_w2aab3b7b4b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] Noboru Endou and Yasunari Shidama. Integral of measurable function. 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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":"2026071213090505446_j_forma-2018-0005_ref_010_w2aab3b7b4b1b6b1ab1ac10Aa","unstructured":"[10] P. R. Halmos. Measure Theory. Springer-Verlag, 1974."},{"key":"2026071213090505446_j_forma-2018-0005_ref_011_w2aab3b7b4b1b6b1ab1ac11Aa","doi-asserted-by":"crossref","unstructured":"[11] Johannes H\u00f6lzl and Armin Heller. Three chapters of measure theory in Isabelle\/HOL. In Marko C. J. D. van Eekelen, Herman Geuvers, Julien Schmaltz, and Freek Wiedijk, editors, Interactive Theorem Proving (ITP 2011), volume 6898 of LNCS, pages 135\u2013151, 2011.10.1007\/978-3-642-22863-6_12","DOI":"10.1007\/978-3-642-22863-6_12"},{"key":"2026071213090505446_j_forma-2018-0005_ref_012_w2aab3b7b4b1b6b1ab1ac12Aa","doi-asserted-by":"crossref","unstructured":"[12] Yasushige Watase, Noboru Endou, and Yasunari Shidama. On L1 space formed by real-valued partial functions. 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