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

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

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