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2016 A characterization of $1$-rectifiable doubling measures with connected supports
Jonas Azzam, Mihalis Mourgoglou
Anal. PDE 9(1): 99-109 (2016). 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.

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Jonas Azzam. Mihalis Mourgoglou. "A characterization of $1$-rectifiable doubling measures with connected supports." Anal. PDE 9 (1) 99 - 109, 2016. https://doi.org/10.2140/apde.2016.9.99

Information

Received: 14 January 2015; Revised: 22 June 2015; Accepted: 11 October 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1332.28007
MathSciNet: MR3461302
Digital Object Identifier: 10.2140/apde.2016.9.99

Subjects:
Primary: 28A75
Secondary: 28A78

Keywords: connected metric spaces , doubling measures , porosity , rectifiability

Rights: Copyright © 2016 Mathematical Sciences Publishers

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