{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T01:35:57Z","timestamp":1777426557241,"version":"3.51.4"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T00:00:00Z","timestamp":1490659200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/legalcode"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,3,28]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p> The purpose of this article is to show Fubini\u2019s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini\u2019s theorem, and prove the relevant proposition. Finally we show Fubini\u2019s theorem on measure.<\/jats:p>","DOI":"10.1515\/forma-2017-0001","type":"journal-article","created":{"date-parts":[[2017,5,19]],"date-time":"2017-05-19T10:01:47Z","timestamp":1495188107000},"page":"1-29","source":"Crossref","is-referenced-by-count":4,"title":["Fubini\u2019s Theorem on Measure"],"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":[[2017,5,11]]},"reference":[{"key":"2021040815432667476_j_forma-2017-0001_ref_001_w2aab2b8c18b1b7b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990."},{"key":"2021040815432667476_j_forma-2017-0001_ref_002_w2aab2b8c18b1b7b1ab1ab2Aa","unstructured":"[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990."},{"key":"2021040815432667476_j_forma-2017-0001_ref_003_w2aab2b8c18b1b7b1ab1ab3Aa","unstructured":"[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990."},{"key":"2021040815432667476_j_forma-2017-0001_ref_004_w2aab2b8c18b1b7b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.","DOI":"10.1515\/9783110866209"},{"key":"2021040815432667476_j_forma-2017-0001_ref_005_w2aab2b8c18b1b7b1ab1ab5Aa","unstructured":"[5] J\u00f3zef Bia\u0142as. The \u03c3-additive measure theory. 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Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004."},{"key":"2021040815432667476_j_forma-2017-0001_ref_019_w2aab2b8c18b1b7b1ab1ac19Aa","unstructured":"[19] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990."},{"key":"2021040815432667476_j_forma-2017-0001_ref_020_w2aab2b8c18b1b7b1ab1ac20Aa","doi-asserted-by":"crossref","unstructured":"[20] 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):231-236, 2007. doi: 10.2478\/v10037-007-0026-3.","DOI":"10.2478\/v10037-007-0026-3"},{"key":"2021040815432667476_j_forma-2017-0001_ref_021_w2aab2b8c18b1b7b1ab1ac21Aa","unstructured":"[21] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Limit of sequence of subsets. 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