A characterization of 1-rectifiable doubling measures with connected supports

Abstract

Garnett, Killip, and Schul have exhibited a doubling measure μ with support equal to Rd that is 1- rectifiable, meaning there are countably many curves Γi of finite length for which μ(Rd\υΓi) = 0. In this note, we characterize when a doubling measure μ with support equal to a connected metric space X has a 1-rectifiable subset of positive measure and show this set coincides up to a set of μ-measure zero with the set of x ∈ X for which lim infr→0 μ(BX (x, r))/r > 0.