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[{"key": "dc.contributor.advisor", "value": "Tuominen, Heli", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Lehtonen, Taru", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-03-24T09:18:47Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-03-24T09:18:47Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1524763", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/49190", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielmassa tarkastellaan ensin Riemannin integraalia ja sen ongelmia rajank\u00e4yntitilanteissa. Suurin ongelma rajank\u00e4ynniss\u00e4 on, ett\u00e4 Riemannintegraalien\njonon raja-arvo ei v\u00e4ltt\u00e4m\u00e4tt\u00e4 aina ole sama kuin rajafunktion\nRiemann-integraali. Lis\u00e4ksi todetaan, ett\u00e4 Riemann-integroituvien funktioiden\njoukko on melko pieni. Seuraavana esitell\u00e4\u00e4n porrasfunktioiden integraali\nominaisuuksineen. T\u00e4m\u00e4n j\u00e4lkeen perehdyt\u00e4\u00e4n Riemann-integroituvien\nfunktioiden luokkaa suurempaan yl\u00e4funktioiden luokkaan L+ ja lis\u00e4ksi osoitetaan,\nett\u00e4 Riemann-integroituvat funktiot kuuluvat yl\u00e4funktioiden luokkaan.\n\nYl\u00e4funktioiden luokan esittelyn j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n Lebesguen integraali\nja perehdyt\u00e4\u00e4n sen ominaisuuksiin. Lebesguen integraali m\u00e4\u00e4ritell\u00e4\u00e4n\nRieszin m\u00e4\u00e4ritelm\u00e4n mukaan, sill\u00e4 se on tiivistetympi, suoraviivaisempi\nja johtaa nopeammin asian ytimeen kuin Lebesguen alkuper\u00e4inen m\u00e4\u00e4ritelm\u00e4.\nLis\u00e4ksi laajennetaan yl\u00e4funktioiden luokka Lebesgue-integroituvien\nfunktioiden luokkaan L ja osoitetaan t\u00e4m\u00e4n olevan selv\u00e4sti suurempi kuin\nyl\u00e4funktioiden luokka.\n\nViimeisess\u00e4 kappaleessa perehdyt\u00e4\u00e4n Lebesguen integraalin rajank\u00e4yntiin\nmonotonisen konvergenssin lauseen ja dominoidun konvergenssin\nlauseen avulla. Dominoidun konvergenssin lause on yksi Lebesguen integraalin\nt\u00e4rkeimmist\u00e4 tuloksista. Tiivistetysti konvergenssilauseiden sanoma on,\nett\u00e4 integroinnin ja rajank\u00e4ynnin j\u00e4rjestyst\u00e4 voidaan vaihtaa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Taru Lehtonen (tapanikk) on 2016-03-24 09:18:47.385343. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-03-24T09:18:47Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201603241939.pdf: 412273 bytes, checksum: 411b28a8d64c6691a683538a59a1365a (MD5)\nlicense.html: 4800 bytes, checksum: 7bf111020fb82a84709324b50ea02482 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-03-24T09:18:47Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201603241939.pdf: 412273 bytes, checksum: 411b28a8d64c6691a683538a59a1365a (MD5)\nlicense.html: 4800 bytes, checksum: 7bf111020fb82a84709324b50ea02482 (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (50 s.)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Lebesguen integraali", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Riemannin integraali", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yl\u00e4funktio", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "konvergenssilause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Lebesguen integraali - Rieszin m\u00e4\u00e4ritelm\u00e4", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201603241939", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2016-03-24T09:18:48Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "integraalilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "konvergenssi", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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