A characterization of 1-rectifiable doubling measures with connected supports

Azzam, Jonas; Mourgoglou, Mihalis

doi:10.2140/apde.2016.9.99

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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 $μ$ with support equal to $ℝ^{d}$ that is $1$ -rectifiable, meaning there are countably many curves $Γ_{i}$ of finite length for which $μ (ℝ^{d} ∖ ⋃ Γ_{i}) = 0$ . 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 ${liminf}_{r \to 0} μ (B_{X} (x, r)) ∕ r > 0$ .

Keywords

doubling measures, rectifiability, porosity, connected metric spaces

Mathematical Subject Classification 2010

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

Vol. 9, No. 1, 2016