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

TL;DR

This paper characterizes when a doubling measure on a connected metric space has a 1-rectifiable subset of positive measure, linking rectifiability to a measure density condition at points.

Contribution

It provides a characterization of 1-rectifiable doubling measures on connected metric spaces, extending previous results to more general settings.

Findings

A doubling measure with support on a connected space has a positive measure 1-rectifiable subset if and only if the measure density condition holds.

The 1-rectifiable set coincides with points where the measure density ratio is positive in the limit.

The characterization generalizes known results from Euclidean spaces to connected metric spaces.

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 $\liminf_{r\rightarrow 0} \mu(B_{X}(x,r))/r>0$.