Calculus of Variations and Geometric Measure Theory

A. Marchese - A. Merlo

Characterization of rectifiability via Lusin type approximation

created by marchese on 30 Dec 2021
modified on 26 Dec 2022


Accepted Paper

Inserted: 30 dec 2021
Last Updated: 26 dec 2022

Journal: Analysis & PDE
Year: 2021


We prove that a Radon measure $\mu$ on $\mathbb{R}^n$ can be written as $\mu=\sum_{i=0}^n\mu_i$, where each of the $\mu_i$ is an $i$-dimensional rectifiable measure if and only if for every Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ and every $\varepsilon>0$ there exists a function $g$ of class $C^1$ such that $\mu(\{x\in\mathbb{R}^n:g(x)\neq f(x)\})<\varepsilon$.