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[{"key": "dc.contributor.advisor", "value": "Lehtonen, Ari", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Ojalehto, Jennika", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-05-31T16:55:24Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-05-31T16:55:24Z", "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:1542539", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/50017", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutustua Jordanin sis\u00e4lt\u00f6\u00f6n ja Lebesguen ulkomittaan reaaliakselin v\u00e4lill\u00e4 ja tason joukossa, joita k\u00e4ytet\u00e4\u00e4n muun muassa tutkittaessa funktion Riemann-\\hskip0pt integroituvuutta.\n Tutkielmassa tutustutaan Jordanin sis\u00e4- ja ulkosis\u00e4ll\u00f6n sek\u00e4 Lebesguen ulkomitan t\u00e4rkeimpiin ominaisuuksiin sek\u00e4 niiden v\u00e4liseen yhteyteen. Lis\u00e4ksi k\u00e4sitell\u00e4\u00e4n Jordanin ja Lebesguen ehdot funktion Riemann-integroituvuudelle.\n\nTutkielman aluksi kerrataan analyysin perusteista reaaliakselin v\u00e4lin Riemannin integraali sek\u00e4 mitta- ja integraaliteorian k\u00e4site nollamittaisuus, jotka ovat tutkielman kannalta t\u00e4rkeit\u00e4 asioita. \nLis\u00e4ksi tutustutaan funktion oskillaatioon eli funktion arvojen heilahteluun reaaliakselin v\u00e4lill\u00e4. \nT\u00e4m\u00e4 on keskeisess\u00e4 asemassa tutkittaessa Riemann-integroituvuutta Jordanin ulkosis\u00e4ll\u00f6n avulla. Jordanin kriteeriss\u00e4 tutkitaan joukkoa, jossa funktion oskillaatio kasvaa suuremmaksi tai on yht\u00e4 suuri kuin annettu luku $\\epsilon$. Funktio on Riemann-integroituva jos ja vain jos t\u00e4m\u00e4 joukko on nollamittainen.\nLis\u00e4ksi tutustutaan Lebesguen ulkomittaan ja sen ominaisuuksiin sek\u00e4 Lebesguen ehtoon Riemann-integroituvuudelle. Lebesguen ehdon mukaan funktio on Riemann-integroituva jos ja vain jos ep\u00e4jatkuvuuspisteiden joukon Lebesguen ulkomitta on nolla.\nEsimerkit ja kuvat havainnollistavat mit\u00e4 hy\u00f6ty\u00e4 Jordanin sis\u00e4- ja ulkosis\u00e4ll\u00f6st\u00e4 sek\u00e4 Lebesguen ulkomitasta on k\u00e4yt\u00e4nn\u00f6ss\u00e4.\n\nTutkielman lopuksi tutustutaan vastaaviin asioihin kuin ensimm\u00e4isess\u00e4 luvussa, mutta reaaliakselin v\u00e4lin sijasta tutkitaan asioita tason joukossa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Jennika Ojalehto (jejupiis) on 2016-05-31 16:55:24.328032. 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 2016-05-31T16:55:24Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201605312792.pdf: 507785 bytes, checksum: 4f2ac103c76d4176dbec2abf1b7a7c96 (MD5)\nlicense.html: 4803 bytes, checksum: 28cf19fdce9176eb42da65426b200923 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-05-31T16:55:24Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201605312792.pdf: 507785 bytes, checksum: 4f2ac103c76d4176dbec2abf1b7a7c96 (MD5)\nlicense.html: 4803 bytes, checksum: 28cf19fdce9176eb42da65426b200923 (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (45 sivua)", "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": "Riemannin integraali", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Jordanin sis\u00e4lt\u00f6 ja Lebesguen ulkomitta", "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-201605312792", "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": "2016-05-31T16:55:25Z", "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": "funktiot", "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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