Calculus of Variations and Geometric Measure Theory

A. Marchese

Lusin type theorems for Radon measures

created by marchese on 23 Nov 2015
modified on 13 Sep 2017


Rend. Semin. Mat. Univ. Padova

Inserted: 23 nov 2015
Last Updated: 13 sep 2017

Year: 2015


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