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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Salo, Henna-Kaisa", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-12T06:15:25Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-12T06:15:25Z", "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/123456789/64539", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on osoittaa, ett\u00e4 Lebesguen mitta on kiertoinvariantti, sek\u00e4 esitell\u00e4 lukijalle Hausdorffin mitta ja sen ominaisuuksia. Lebesguen mitan kiertoinvarianttisuuden todistus vaatii paljon esitietoja, joita k\u00e4sittelemme ensimm\u00e4isess\u00e4 luvussa. Toisessa luvussa todistamme, ett\u00e4 Lebesguen mitta on kiertoinvariantti, ja kolmannessa luvussa tutustumme Hausdorffin mittaan. \n\nEnsimm\u00e4isess\u00e4 luvussa tarkastelemme useita Mitta- ja integraaliteorian kurssilta opiskelijalle tuttuja m\u00e4\u00e4ritelmi\u00e4 ja lauseita, esimerkiksi Lebesguen ulkomittaan, Lebesgue-mitallisuuteen ja lopuksi my\u00f6s Lebesguen mittaan liittyen. Er\u00e4it\u00e4 t\u00e4rkeit\u00e4 tuloksia ovat mitallisten joukkojen ominaisuudet ja Lebesguen mitan siirtoinvarianttisuus. N\u00e4it\u00e4 tuloksia tarvitsemme my\u00f6hemmin, jotta voimme osoittaa Lebesguen mitan olevan kiertoinvariantti.\n\nToisen luvun alussa m\u00e4\u00e4rittelemme Whitney-kuutiot, jotka ovat puoliavoimia kuutioita, joiden sivut ovat koordinaattiakselien suuntaiset. Esittelemme my\u00f6s muita Whitney-kuutioiden ominaisuuksia. T\u00e4m\u00e4n j\u00e4lkeen kertaamme k\u00e4\u00e4ntyv\u00e4n lineaarikuvauksen ominaisuuksia ja m\u00e4\u00e4rittelemme kiertokuvauksen. N\u00e4iden valmistelujen j\u00e4lkeen olemme valmiit todistamaan tutkielman p\u00e4\u00e4tuloksen: mille tahansa avaruuden R^n osajoukolle E p\u00e4tee m*(TE)=|det T|m*(E) aina, kun T on k\u00e4\u00e4ntyv\u00e4 lineaarikuvaus. Todistuksen j\u00e4lkeen pystymme toteamaan, ett\u00e4 Lebesguen mitta on kiertoinvariantti, sill\u00e4 kiertokuvaukselle p\u00e4tee |det T|=1.\n\nViimeisess\u00e4 luvussa tutkimme yleist\u00e4 ulkomittaa \u03bc ja osoitamme, ett\u00e4 \u03bc-mitallisten joukkojen kokoelma muodostaa \u03c3-algebran. Osoitamme my\u00f6s, ett\u00e4 metrinen ulkomitta on Borel-mitta. T\u00e4m\u00e4n j\u00e4lkeen tutustumme Hausdorffin mittaan, ja osoitamme sen olevan ulkomitta, Borel-mitta ja Borel-s\u00e4\u00e4nn\u00f6llinen. Lopuksi toteamme, ett\u00e4 Hausdorffin ja Lebesguen n-ulotteiset mitat ovat yht\u00e4 suuret avaruudessa R^n.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-06-12T06:15:25Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-12T06:15:25Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "33", "language": "", "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": "kiertoinvarianttisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Hausdorffin mitta", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Lebesguen mitan kiertoinvarianttisuus ja Hausdorffin mitta", "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-201906123141", "language": "", "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": "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": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in 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.subject.yso", "value": "matematiikka", "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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