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 |