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Doubling measures with doubling continuous part


Authors: Man-Li Lou and Min Wu
Journal: Proc. Amer. Math. Soc. 138 (2010), 3585-3589
MSC (2010): Primary 28C15
DOI: https://doi.org/10.1090/S0002-9939-10-10358-X
Published electronically: April 13, 2010
MathSciNet review: 2661557
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Abstract: We prove that every compact subset of $ \mathbb{R}^d$ of positive Lebesgue measure carries a doubling measure which is not purely atomic. Also, we prove that for every compact and nowhere dense subset $ E$ of  $ \mathbb{R}^d$ without isolated points and for every doubling measure $ \mu$ on $ E$ there is a countable set $ F$ with $ E\cap F=\emptyset$ and a doubling measure $ \nu$ on $ E\cup F$ such that $ \nu\vert _E=\mu$. This shows that there are many doubling measures whose continuous part is doubling.


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Additional Information

Man-Li Lou
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
Email: loumanli@126.com

Min Wu
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China
Email: wumin@scut.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-10-10358-X
Keywords: doubling measure, purely atomic, continuous part
Received by editor(s): November 16, 2009
Received by editor(s) in revised form: December 30, 2009
Published electronically: April 13, 2010
Additional Notes: This work was supported by National Natural Science Foundation of China (Grants No. 10571063, 10631040)
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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