Two sufficient conditions for rectifiable measures

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 $\hat{I}¼$ on ${\mathbb{R}}^n$ satisfying the global regularity hypothesis $$\underset{r\downarrow 0}{\mathrm{limsup}}\hat{I}¼(B(x,r))/r^m<\infty \text{at I^¼-a.e. x∈Rn}$$ 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 $\hat{I}¼$ on ${\mathbb{R}}^n$ with $$\underset{r\downarrow 0}{\lim }\hat{I}¼(B(x,r))/r=\infty \text{at I^¼-a.e. x∈Rn}$$ is 1-rectifiable.

10 pages (v2: updated statement of Corollary 1.12, minor corrections and new reference added)