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 μ with support equal to d that is 1-rectifiable, meaning there are countably many curves Γi of finite length for which μd Γi = 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 X for which liminf r0μ(BX(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