2005-11-02Buch DOI: 10.18452/2561
Transformation of Lebesgue Measure and Integral by Lipschitz Mappings
 dc.contributor.author Naumann, Joachim dc.date.accessioned 2017-06-15T17:33:30Z dc.date.available 2017-06-15T17:33:30Z dc.date.created 2005-11-02 dc.date.issued 2005-11-02 dc.identifier.issn 0863-0976 dc.identifier.uri http://edoc.hu-berlin.de/18452/3213 dc.description.abstract We 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.iso eng dc.publisher Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ dc.subject Implicit function theorems eng dc.subject Jacobians eng dc.subject transformations with several variables eng dc.subject None of the above eng dc.subject but in this section eng dc.subject.ddc 510 Mathematik dc.title Transformation of Lebesgue Measure and Integral by Lipschitz Mappings dc.type book dc.identifier.urn urn:nbn:de:kobv:11-10051810 dc.identifier.doi http://dx.doi.org/10.18452/2561 local.edoc.pages 30 local.edoc.type-name Buch local.edoc.container-type series local.edoc.container-type-name Schriftenreihe local.edoc.container-year 2005 dc.identifier.zdb 2075199-0 bua.series.name Preprints aus dem Institut für Mathematik bua.series.issuenumber 2005,8