Characterization of rectifiability via Lusin-type approximation

Abstract

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