Characterization of rectifiable measures in terms of α-numbers

2018/08/23 by Jonas Azzam, Xavier Tolsa, Azzam, Jonas +3
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #28A75 #28A78 #42B20 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Computability, Logic, AI Algorithms #FOS: Mathematics #Metric Geometry (math.MG) #Risk and Portfolio Optimization #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1808.07661

openalex publication_date 2018/08/23 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

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

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