Characterization of rectifiable measures in terms of $\alpha$-numbers

Abstract

We characterize Radon measures $\mu$ in ${\mathbb{R}}^n$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\mu \ll {\mathcal{H}}^d$. The characterization is in terms of a Jones function involving the so-called $\alpha$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.