## 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

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

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## Additional Information

**Matthew Badger**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- MR Author ID: 962755
- 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
- 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: https://doi.org/10.1090/proc/12881
- MathSciNet review: 3477060