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

TL;DR

This paper provides a new characterization of $d$-rectifiable Radon measures in Euclidean space using a Jones function based on $eta$-numbers, answering an open question in geometric measure theory.

Contribution

It introduces a characterization of rectifiable measures via a Jones function involving $eta$-numbers, resolving an open problem from previous research.

Findings

Characterization of rectifiable measures using $eta$-numbers.

Answer to an open question in geometric measure theory.

Connection between measure rectifiability and Jones functions.

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.