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Two sufficient conditions for rectifiable measures


Authors: Matthew Badger and Raanan Schul
Journal: Proc. Amer. Math. Soc. 144 (2016), 2445-2454
MSC (2010): Primary 28A75
DOI: https://doi.org/10.1090/proc/12881
Published electronically: October 5, 2015
MathSciNet review: 3477060
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Abstract: We identify two sufficient conditions for locally finite Borel measures on $ \mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $ \mathbb{R}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $ L^p$ affine approximability for a locally finite Borel measure $ \mu $ on $ \mathbb{R}^n$ satisfying the global regularity hypothesis

$\displaystyle \limsup _{r\downarrow 0} \mu (B(x,r))/r^m <\infty$$\displaystyle \quad \text {at $\mu $-a.e.~$x\in \mathbb{R}^n$}$

to be $ m$-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $ \mu $ on $ \mathbb{R}^n$ with

$\displaystyle \lim _{r\downarrow 0} \mu (B(x,r))/r=\infty$$\displaystyle \quad \text {at $\mu $-a.e.~$x\in \mathbb{R}^n$}$

is 1-rectifiable.

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Additional Information

Matthew Badger
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: matthew.badger@uconn.edu

Raanan Schul
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: schul@math.sunysb.edu

DOI: https://doi.org/10.1090/proc/12881
Keywords: Rectifiable measure, singular measure, Jones beta number, Hausdorff density, Hausdorff measure
Received by editor(s): January 7, 2015
Received by editor(s) in revised form: June 30, 2015
Published electronically: October 5, 2015
Additional Notes: The first author was partially supported by an NSF postdoctoral fellowship DMS 1203497
The second author was partially supported by NSF DMS 1361473
Communicated by: Tatiana Toro
Article copyright: © Copyright 2015 American Mathematical Society