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 ${\mathrm{liminf}}_{r\to 0}\mu (B_X(x,r))/r>0$.

Made suggested referee changes. Published in Anal. PDE 9 (2016), no. 1, 99--109

References in corpus (1)

• Multiscale analysis of 1-rectifiable measures: necessary conditions

Cited by in corpus (3)

• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability
• Sufficient condition for rectifiability involving Wasserstein distance $W_2$