Characterization of rectifiability via Lusin type approximation

Abstract

We prove that a Radon measure $\hat{I}¼$ on ${\mathbb{R}}^n$ can be written as $\hat{I}¼={\sum }_{i=0}^n\hat{I}{¼}_i$, where each of the $\hat{I}{¼}_i$ is an $i$-dimensional rectifiable measure if and only if for every Lipschitz function $f:{\mathbb{R}}^n\to \mathbb{R}$ and every $\epsilon >0$ there exists a function $g$ of class $C^1$ such that $\hat{I}¼(\{x\in {\mathbb{R}}^n:g(x)\ne f(x)\})<\epsilon$.