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2005-11-02Buch DOI: 10.18452/2561
Transformation of Lebesgue Measure and Integral by Lipschitz Mappings
dc.contributor.authorNaumann, Joachim
dc.date.accessioned2017-06-15T17:33:30Z
dc.date.available2017-06-15T17:33:30Z
dc.date.created2005-11-02
dc.date.issued2005-11-02
dc.identifier.issn0863-0976
dc.identifier.urihttp://edoc.hu-berlin.de/18452/3213
dc.description.abstractWe first show that {\sc Lipschitz} mappings transform measurable sets into measurable sets. Then we prove the following theorem:\par {\it Let $E\subseteq \mathbb{R}^n$ be open, and let $\phi: E\to\mathbb{R}^n$ be continuous. If $\phi$ is differentiable at $x_0\in E$, then} \[ \lim\limits_{r\to 0} \dfrac{\lambda_n(\phi(B_r(x_0)))}{\lambda_n(B_r)} = \Big| \det\phi'(x_0)\Big|. \] From this result the change of variables formula for injective and locally {\sc Lipschitz} mappings is easily derived by using the {\sc Radon-Nikodym} theorem. We finally discuss the transformation of $L^p$ functions by {\sc Lipschitz} mappings.eng
dc.language.isoeng
dc.publisherHumboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/
dc.subjectImplicit function theoremseng
dc.subjectJacobianseng
dc.subjecttransformations with several variableseng
dc.subjectNone of the aboveeng
dc.subjectbut in this sectioneng
dc.subject.ddc510 Mathematik
dc.titleTransformation of Lebesgue Measure and Integral by Lipschitz Mappings
dc.typebook
dc.identifier.urnurn:nbn:de:kobv:11-10051810
dc.identifier.doihttp://dx.doi.org/10.18452/2561
local.edoc.pages30
local.edoc.type-nameBuch
local.edoc.container-typeseries
local.edoc.container-type-nameSchriftenreihe
local.edoc.container-year2005
dc.identifier.zdb2075199-0
bua.series.namePreprints aus dem Institut für Mathematik
bua.series.issuenumber2005,8

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