#### Vol. 9, No. 1, 2016

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

### Jonas Azzam and Mihalis Mourgoglou

Vol. 9 (2016), No. 1, 99–109
DOI: 10.2140/apde.2016.9.99
##### 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 ${liminf}_{r\to 0}\mu \left({B}_{X}\left(x,r\right)\right)∕r>0$.

##### Keywords
doubling measures, rectifiability, porosity, connected metric spaces
Primary: 28A75
Secondary: 28A78
##### Milestones
Received: 14 January 2015
Revised: 22 June 2015
Accepted: 11 October 2015
Published: 10 February 2016
##### Authors
 Jonas Azzam Departament de Matemàtiques Universitat Autònoma de Barcelona Edifici C Facultat de Ciències 08193 Bellaterra Spain Mihalis Mourgoglou Departament de Matemàtiques Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica Edifici C Facultat de Ciències 08193 Bellaterra Spain