We say that a Radon measure \(\mu\) on \(\mathbb{R}^d\) is \(n\)-rectifiable (for \(1\leq n\leq d\)) if it is absolutely continuous with respect to \(n\)-dimensional Hausdorff measure \(\mathcal{H}^n\)and there exist countable many Lipschitz maps \(f_i\colon \mathbb{R}^n\to \mathbb{R}^d\) such that a complement of \(\bigcup_i f_i(\mathbb{R}^n)\) to \(\mathbb{R}^n\) is a set of \(\mu\)-measure zero.
Take \(1\leq p<\infty\) and two probability Borel measures \(\mu\), \(\nu\) on \(\mathbb{R}^d\) such that \(\int|x|^p d\mu\) and \(\int|x|^p d\nu\) are finite. The Wasserstein distance between \(\mu\) and \(\nu\) is defined as a number \[ W_p(\mu, \nu)=\left( \inf_{\pi} \int_{\mathbb{R}^d\times \mathbb{R}^d}|x-y|^p d\pi(x,y)\right)^{1/p}, \] where \(\pi\) are the Borel probability measures on \(\mathbb{R}^d\times \mathbb{R}^d\) such that \(\pi(A\times \mathbb{R}^d)=\mu(A)\) and \(\pi(\mathbb{R}^d \times A)=\nu(A)\) for all measurable sets \(A\subset \mathbb{R}^d\). It is a way to measure the cost of transporting one measure to another and to characterize optimal transport.
The main theorem of the paper is a necessary condition for rectifiability of measures, which states that if \(\mu\) is \(n\)-rectifiable on \(\mathbb{R}^d\) than for \(\mu\)-a.e. \(x\in \mathbb{R}^d\) the following inequality holds: \(\int\limits_0^1\alpha_{\mu,2}(x,r)^2 \frac{\text{d}r}{r}<\infty\), where \(\alpha_{\mu,2}(x,r)\) are the numbers defined with the use of Wasserstein distance. Since it was earlier proved that it is also a sufficient condition for rectifiability, the author obtains the characterization of this fact.