Calculus of Variations and Geometric Measure Theory
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A. Marchese

Lusin type theorems for Radon measures

created by marchese on 23 Nov 2015
modified on 24 May 2016

[BibTeX]

Preprint

Inserted: 23 nov 2015
Last Updated: 24 may 2016

Year: 2015

Abstract:

We add to the literature the following observation. If $\mu$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $\mu$-a.e. then for every $C^1$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ there holds $\mu\{x\in\mathbb{R}^n: f(x)=g(x)\}=0.$ In other words the Lusin type approximation property of Lipschitz functions with $C^1$ functions does not hold with respect to a general Radon measure.

Keywords: Lipschitz functions, Lusin type approximation, Radon measure, Porous set


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