Characterization of rectifiability via Lusin type approximation

TL;DR

This paper characterizes rectifiable measures in Euclidean space through a Lusin-type approximation property, showing that measures decomposable into rectifiable parts can be approximated by smooth functions on large measure sets.

Contribution

It establishes a new equivalence between rectifiability of measures and the existence of smooth approximations for Lipschitz functions, providing a novel characterization of rectifiable measures.

Findings

Measures decomposable into rectifiable parts admit smooth approximations of Lipschitz functions.

The approximation property characterizes the rectifiability of measures.

The result links geometric measure theory with smooth approximation techniques.

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