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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Two sufficient conditions for rectifiable measures
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by Matthew Badger and Raanan Schul PDF
Proc. Amer. Math. Soc. 144 (2016), 2445-2454 Request permission


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 \[ \limsup _{r\downarrow 0} \mu (B(x,r))/r^m <\infty \;\; \text {at $\mu $-a.e.~$x\in \mathbb {R}^n$} \] 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 \[ \lim _{r\downarrow 0} \mu (B(x,r))/r=\infty \;\; \text {at $\mu $-a.e.~$x\in \mathbb {R}^n$}\] is 1-rectifiable.
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Additional Information
  • Matthew Badger
  • Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
  • MR Author ID: 962755
  • Email:
  • Raanan Schul
  • Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
  • Email:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 2445-2454
  • MSC (2010): Primary 28A75
  • DOI:
  • MathSciNet review: 3477060