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 $f:{\mathbb{R}}^n\to \mathbb{R}$ is a Lipschitz function which is non-differentiable $\mu$-a.e. then for every $C^1$ function $g:{\mathbb{R}}^n\to \mathbb{R}$ it 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.