Lusin type theorems for Radon measures

TL;DR

This paper demonstrates that for certain singular measures on b^n, Lipschitz functions cannot be approximated by smooth functions on sets of positive measure, highlighting limitations of Lusin-type theorems in this context.

Contribution

It establishes that Lusin approximation properties fail for Lipschitz functions with respect to measures that assign zero measure to porous sets and are singular.

Findings

Lusin approximation does not hold for Lipschitz functions under these measures.

Non-differentiability of Lipschitz functions is generic c-measure.

The result applies to measures assigning zero measure to porous sets.

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 $g:\mathbb{R}^n\rightarrow\mathbb{R}$ 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.