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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Auvinen, Niilo", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-03-04T09:04:51Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-03-04T09:04:51Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/63003", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa perehdyt\u00e4\u00e4n merkkimittoihin ja niihin liittyviin hajotelmalauseisiin. Lis\u00e4ksi p\u00e4\u00e4lauseena todistetaan mittateorian perustuloksiin kuuluva Radonin ja Nikodymin lause kolmessa eri tilanteessa: ensin kahden \u00e4\u00e4rellisen mitan tapauksessa, sitten sigma-\u00e4\u00e4rellisten mittojen kanssa ja viimeisen\u00e4 sigma-\u00e4\u00e4rellisen mitan ja merkkimitan tapauksessa.\n\nMerkkimitat ovat mittateoriassa tutkittuja mittojen yleistyksi\u00e4. Ne voivat mitoista poiketen saada my\u00f6s negatiivisia arvoja, mutta kuitenkin niin, ettei merkkimitta voi saavuttaa sek\u00e4 positiivista ett\u00e4 negatiivista \u00e4\u00e4ret\u00f6nt\u00e4. Tutkielmassa tutustutaan merkkimittojen absoluuttiseen jatkuvuuteen ja keskin\u00e4iseen singulaarisuuteen. Ensimm\u00e4inen viittaa merkkimittojen vahvaan riippuvuuteen toisistaan: joukon nollamittaisuus periytyy my\u00f6s toiselle merkkimitalle. Singulaarisuus taas p\u00e4invastoin kertoo joukkofunktioiden t\u00e4ydellisest\u00e4 riippumattomuudesta: ne saavat nollasta poikkeavia arvoja t\u00e4ysin eri osissa avaruutta.\n\nTutkielmassa todistetaan kolme hajotelmalausetta. Hahnin hajotelmalauseen nojalla mitta-avaruus voidaan jakaa merkkimitan suhteen kahteen pistevieraaseen osaan, joista toisessa merkkimitta saa vain positiivisia arvoja ja toisessa taas vain negatiivisia arvoja. Kyseisell\u00e4 lauseella on oleellinen rooli Radonin ja Nikodymin lauseen todistuksessa. Jordanin hajotelmalauseessa todistetaan, miten jokainen merkkimitta voidaan palauttaa kahden mitan erotukseksi, ja viimeisen\u00e4 Lebesguen hajotelmalause osoittaa, ett\u00e4 kahta merkkimittaa tutkittaessa kumpi tahansa voidaan hajottaa toisen suhteen absoluuttisesti jatkuvaan ja singulaariseen osaan.\n\nOn helppoa osoittaa, ett\u00e4 mitallista ei-negatiivista funktiota integroimalla saadaan luotua mitta. Ei ole my\u00f6sk\u00e4\u00e4n haastavaa n\u00e4ytt\u00e4\u00e4, ett\u00e4 n\u00e4in saatu mitta on absoluuttisesti jatkuva integroinnissa k\u00e4ytetyn mitan suhteen. Radonin ja Nikodymin lause todistaa, ett\u00e4 sama p\u00e4tee tietyill\u00e4 lis\u00e4oletuksilla my\u00f6s k\u00e4\u00e4nteisesti: Jos sigma-\u00e4\u00e4rellinen (merkki)mitta v on absoluuttisesti jatkuva sigma-\u00e4\u00e4rellisen mitan m suhteen, on olemassa mitallinen funktio f, jolle p\u00e4tee, ett\u00e4 jokaisen mitallisen joukon E v-mitta on t\u00e4sm\u00e4lleen funktion f integraali mitan m suhteen joukon E yli. K\u00e4y siis ilmi, ett\u00e4 sigma-\u00e4\u00e4rellisten mittojen tapauksessa absoluuttinen jatkuvuus voidaan karakterisoida t\u00e4ysin mitallisten funktioiden integroinniksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-03-04T09:04:51Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-03-04T09:04:51Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "46", "language": "", "element": "format", "qualifier": "extent", "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": "integraaliteoria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "merkkimitta", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "absoluuttinen jatkuvuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Radonin ja Nikodymin lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Radonin ja Nikodymin lause", "language": "", "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-201903041706", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "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": "yvv.contractresearch.funding", "value": "0", "language": "", "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": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "mittateoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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