A characterization of $1$-rectifiable doubling measures with connected supports

Abstract

Garnett, Killip, and Schul have exhibited a doubling measure $\mu$ with support equal to $\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $\Gamma_{i}$ of finite length for which $\mu(\mathbb{R}^{d}\backslash \bigcup \Gamma_{i})=0$. In this note, we characterize when a doubling measure $\mu$ 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 $\mu$-measure zero with the set of $x\in X$ for which $\liminf_{r\rightarrow 0} \mu(B_{X}(x,r))/r>0$.