{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T02:59:07Z","timestamp":1648522747764},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,1,1]]},"abstract":"<jats:title>Lebesgue's Convergence Theorem of Complex-Valued Function<\/jats:title>\n        <jats:p>In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.<\/jats:p>","DOI":"10.2478\/v10037-009-0015-9","type":"journal-article","created":{"date-parts":[[2009,7,15]],"date-time":"2009-07-15T01:04:59Z","timestamp":1247619899000},"page":"137-145","source":"Crossref","is-referenced-by-count":1,"title":["Lebesgue's Convergence Theorem of Complex-Valued Function"],"prefix":"10.2478","volume":"17","author":[{"given":"Keiko","family":"Narita","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Noboru","family":"Endou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"key":"1","first-page":"91","article-title":"The ordinal numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"key":"2","first-page":"173","article-title":"Series of positive real numbers. Measure theory","volume":"1","author":"J\u00f3zef Bia\u0142as","year":"1991","journal-title":"Formalized Mathematics"},{"key":"3","first-page":"263","article-title":"The \u03c3-additive measure theory","volume":"2","author":"J\u00f3zef Bia\u0142as","year":"1991","journal-title":"Formalized Mathematics"},{"key":"4","first-page":"507","article-title":"The complex numbers","volume":"3","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"key":"5","first-page":"55","article-title":"Functions and their basic properties","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"key":"6","first-page":"153","article-title":"Functions from a set to a set","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"key":"7","first-page":"357","article-title":"Partial functions","volume":"2","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"key":"8","doi-asserted-by":"publisher","first-page":"167","DOI":"10.2478\/v10037-008-0023-1","article-title":"The Lebesgue monotone convergence theorem","volume":"2","author":"Noboru Endou","year":"2008","journal-title":"Formalized Mathematics"},{"key":"9","doi-asserted-by":"publisher","first-page":"53","DOI":"10.2478\/v10037-006-0008-x","article-title":"Integral of measurable function","volume":"2","author":"Noboru Endou","year":"2006","journal-title":"Formalized Mathematics"},{"key":"10","doi-asserted-by":"publisher","first-page":"57","DOI":"10.2478\/v10037-008-0009-z","article-title":"Egoroff's theorem","volume":"1","author":"Noboru Endou","year":"2008","journal-title":"Formalized Mathematics"},{"key":"11","first-page":"495","article-title":"Definitions and basic properties of measurable functions","volume":"3","author":"Noboru Endou","year":"2001","journal-title":"Formalized Mathematics"},{"key":"12","first-page":"35","article-title":"Basic properties of real numbers","volume":"1","author":"Krzysztof Hryniewiecki","year":"1990","journal-title":"Formalized Mathematics"},{"key":"13","first-page":"273","article-title":"Convergent sequences and the limit of sequences","volume":"2","author":"Jaros\u0142aw Kotowicz","year":"1990","journal-title":"Formalized Mathematics"},{"key":"14","first-page":"269","article-title":"Real sequences and basic operations on them","volume":"2","author":"Jaros\u0142aw Kotowicz","year":"1990","journal-title":"Formalized Mathematics"},{"key":"15","doi-asserted-by":"publisher","first-page":"319","DOI":"10.2478\/v10037-008-0039-6","article-title":"Integral of complex-valued measurable function","volume":"4","author":"Keiko Narita","year":"2008","journal-title":"Formalized Mathematics"},{"key":"16","doi-asserted-by":"publisher","first-page":"89","DOI":"10.2478\/v10037-009-0010-1","article-title":"The measurability of complex-valued functional sequences","volume":"2","author":"Keiko Narita","year":"2009","journal-title":"Formalized Mathematics"},{"key":"17","first-page":"265","article-title":"Conjugate sequences, bounded complex sequences and convergent complex sequences","volume":"2","author":"Adam Naumowicz","year":"1997","journal-title":"Formalized Mathematics"},{"key":"18","first-page":"401","article-title":"\u03c3-fields and probability","volume":"2","author":"Andrzej N\u0119dzusiak","year":"1990","journal-title":"Formalized Mathematics"},{"key":"19","first-page":"147","article-title":"Families of sets","volume":"1","author":"Beata Padlewska","year":"1990","journal-title":"Formalized Mathematics"},{"key":"20","first-page":"17","article-title":"Functional sequence from a domain to a domain","volume":"1","author":"Beata Perkowska","year":"1992","journal-title":"Formalized Mathematics"},{"key":"21","first-page":"449","article-title":"Series","volume":"4","author":"Konrad Raczkowski","year":"1991","journal-title":"Formalized Mathematics"},{"key":"22","doi-asserted-by":"publisher","first-page":"143","DOI":"10.2478\/v10037-006-0018-8","article-title":"Integral of real-valued measurable function","volume":"4","author":"Yasunari Shidama","year":"2006","journal-title":"Formalized Mathematics"},{"key":"23","first-page":"403","article-title":"Convergence and the limit of complex sequences. Series","volume":"3","author":"Yasunari Shidama","year":"1997","journal-title":"Formalized Mathematics"},{"key":"24","first-page":"67","article-title":"Properties of subsets","volume":"1","author":"Zinaida Trybulec","year":"1990","journal-title":"Formalized Mathematics"},{"key":"25","first-page":"73","article-title":"Relations and their basic properties","volume":"1","author":"Edmund Woronowicz","year":"1990","journal-title":"Formalized Mathematics"},{"key":"26","first-page":"181","article-title":"Relations defined on sets","volume":"1","author":"Edmund Woronowicz","year":"1990","journal-title":"Formalized Mathematics"}],"container-title":["Formalized Mathematics"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/17\/2\/article-p137.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/forma.2009.17.issue-2\/v10037-009-0015-9\/v10037-009-0015-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T15:27:45Z","timestamp":1590852465000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10037-009-0015-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,1]]},"references-count":26,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.2478\/v10037-009-0015-9","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2009,1,1]]}}}