Characterization of rectifiable measures in terms of 𝛼-numbers

Abstract

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