Lusin type theorems for Radon measures

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 $f\colon \mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $\mu$-a.e., then for every $C^1$ function $g\colon \mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$ \mu{x\in\mathbb{R}^n\colon f(x)=g(x)}=0. $$ 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.