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 $\mu$ on $\mathbb {R}^n$ satisfying the global regularity hypothesis \[ \limsup _{r\downarrow 0} \mu (B(x,r))/r^m <\infty \;\; \text {at $\mu $-a.e.~$x\in \mathbb {R}^n$} \] 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 \[ \lim _{r\downarrow 0} \mu (B(x,r))/r=\infty \;\; \text {at $\mu $-a.e.~$x\in \mathbb {R}^n$}\] is 1-rectifiable.