{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T11:05:06Z","timestamp":1774868706270,"version":"3.50.1"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T00:00:00Z","timestamp":1756684800000},"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":[[2025,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article, we prove the integrability of continuous functions on\n                    <jats:italic>n<\/jats:italic>\n                    -dimensional real normed spaces, using the Mizar formalism. Generalizing selected theorems from the Mizar Mathematical Library, we prove the integrability of continuous real\n                    <jats:italic>n<\/jats:italic>\n                    -variable functions and then, using the correspondence between product-type and tuple-type spaces, we demonstrate the integrability of continuous functions on the desired multidimensional spaces.\n                  <\/jats:p>","DOI":"10.2478\/forma-2025-0015","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T16:26:31Z","timestamp":1767975991000},"page":"185-206","source":"Crossref","is-referenced-by-count":0,"title":["Integrability of Multivariable Continuous Functions"],"prefix":"10.2478","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5922-2332","authenticated-orcid":false,"given":"Noboru","family":"Endou","sequence":"first","affiliation":[{"name":"National Institute of Technology, Gifu College , Kamimakuwa, Motosu , Gifu , Japan"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Karuizawa Hotch 244-1 , Nagano , Japan"}]}],"member":"374","published-online":{"date-parts":[[2025,12,31]]},"reference":[{"key":"2026033010152277753_j_forma-2025-0015_ref_001","unstructured":"Charalambos D. Aliprantis and Kim C. Border. Infinite dimensional analysis. Springer-Verlag, Berlin, Heidelberg, 2006."},{"key":"2026033010152277753_j_forma-2025-0015_ref_002","unstructured":"Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007."},{"key":"2026033010152277753_j_forma-2025-0015_ref_003","doi-asserted-by":"crossref","unstructured":"Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Improving real analysis in Coq: A user-friendly approach to integrals and derivatives. In Chris Hawblitzel and Dale Miller, editors, Certified Programs and Proofs \u2013 Second International Conference, CPP 2012, Kyoto, Japan, December 13\u201315, 2012. Proceedings, volume 7679 of Lecture Notes in Computer Science, pages 289\u2013304. Springer, 2012. doi:10.1007\/978-3-642-35308-622.","DOI":"10.1007\/978-3-642-35308-6_22"},{"key":"2026033010152277753_j_forma-2025-0015_ref_004","doi-asserted-by":"crossref","unstructured":"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":"2026033010152277753_j_forma-2025-0015_ref_005","doi-asserted-by":"crossref","unstructured":"Noboru Endou. Relationship between the Riemann and Lebesgue integrals. Formalized Mathematics, 29(4):185\u2013199, 2021. doi:10.2478\/forma-2021-0018.","DOI":"10.2478\/forma-2021-0018"},{"key":"2026033010152277753_j_forma-2025-0015_ref_006","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Multidimensional measure space and integration. Formalized Mathematics, 31(1):181\u2013192, 2023. doi:10.2478\/forma-2023-0017.","DOI":"10.2478\/forma-2023-0017"},{"key":"2026033010152277753_j_forma-2025-0015_ref_007","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Universality of measure space. Formalized Mathematics, 32(1):149\u2013163, 2024. doi:10.2478\/forma-2024-0012.","DOI":"10.2478\/forma-2024-0012"},{"key":"2026033010152277753_j_forma-2025-0015_ref_008","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Measure for product space of real normed spaces. Formalized Mathematics, 33(1):117\u2013135, 2025. doi:10.2478\/forma-2025-0010.","DOI":"10.2478\/forma-2025-0010"},{"key":"2026033010152277753_j_forma-2025-0015_ref_009","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Integral of continuous functions of two variables. Formalized Mathematics, 31(1):309\u2013324, 2023. doi:10.2478\/forma-2023-0025.","DOI":"10.2478\/forma-2023-0025"},{"key":"2026033010152277753_j_forma-2025-0015_ref_010","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Integral of continuous three variable functions. Formalized Mathematics, 32(1):9\u201331, 2024. doi:10.2478\/forma-2024-0002.","DOI":"10.2478\/forma-2024-0002"},{"key":"2026033010152277753_j_forma-2025-0015_ref_011","unstructured":"Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from \u211d to \u211d and integrability for continuous functions. Formalized Mathematics, 9(2):281\u2013284, 2001."},{"key":"2026033010152277753_j_forma-2025-0015_ref_012","doi-asserted-by":"crossref","unstructured":"Adam Grabowski and Christoph Schwarzweller. Revisions as an essential tool to maintain mathematical repositories. In M. Kauers, M. Kerber, R. Miner, and W. Windsteiger, editors, Towards Mechanized Mathematical Assistants. Lecture Notes in Computer Science, volume 4573, pages 235\u2013249. Springer: Berlin, Heidelberg, 2007.","DOI":"10.1007\/978-3-540-73086-6_20"},{"key":"2026033010152277753_j_forma-2025-0015_ref_013","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026033010152277753_j_forma-2025-0015_ref_014","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1007\/978-3-642-22863-6_12"},{"key":"2026033010152277753_j_forma-2025-0015_ref_015","unstructured":"M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004."},{"key":"2026033010152277753_j_forma-2025-0015_ref_016","doi-asserted-by":"crossref","unstructured":"Colin Rothgang, Artur Korni\u0142owicz, and Florian Rabe. A new export of the Mizar Mathematical Library. In Fairouz Kamareddine and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics, pages 205\u2013210, Cham, 2021. Springer International Publishing. doi:10.1007\/978-3-030-81097-917.","DOI":"10.1007\/978-3-030-81097-9_17"},{"key":"2026033010152277753_j_forma-2025-0015_ref_017","unstructured":"Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.sciendo.com\/pdf\/10.2478\/forma-2025-0015","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T10:16:04Z","timestamp":1774865764000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.2478\/forma-2025-0015"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,1]]},"references-count":17,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12,31]]},"published-print":{"date-parts":[[2025,9,1]]}},"alternative-id":["10.2478\/forma-2025-0015"],"URL":"https:\/\/doi.org\/10.2478\/forma-2025-0015","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,1]]}}}