Two sufficient conditions for rectifiable measures

TL;DR

This paper establishes two new sufficient conditions involving affine approximability and growth rates of measures that guarantee rectifiability of measures in Euclidean space, extending previous results and providing criteria for different rectifiability cases.

Contribution

It introduces two novel sufficient conditions for rectifiability of measures based on affine approximability and growth rate assumptions, generalizing prior rectifiability criteria.

Findings

First condition extends Pajot's result using $L^p$ affine approximability.

Second condition links growth rate of 1-density to 1-rectifiability.

Provides criteria for measures to be supported on countable Lipschitz images.

Abstract

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ satisfying the global regularity hypothesis $$\limsup_{r\downarrow 0} \mu(B(x,r))/r^m <\infty\quad \text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ to be $m$-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ with $$\lim_{r\downarrow 0} \mu(B(x,r))/r=\infty\quad\text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ is 1-rectifiable.