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 ${ℝ}^d$ that is $1$-rectifiable, meaning there are countably many curves ${\Gamma }_i$ of finite length for which $\mu \left(\right.{ℝ}^d\setminus \bigcup {\Gamma }_i\left)\right.=0$. 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 ${liminf}_{r\to 0}\mu \left(B_X\left(x,r\right)\right)∕r>0$.