# Characterization of rectifiability via Lusin type approximation

created by marchese on 30 Dec 2021
modified on 31 Dec 2021

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Submitted Paper

Inserted: 30 dec 2021
Last Updated: 31 dec 2021

Year: 2021

Abstract:

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 $\mu(\{x\in\mathbb{R}^n:g(x)\neq f(x)\})<\varepsilon$.