Doubling measures with doubling continuous part
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- by Man-Li Lou and Min Wu PDF
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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 $\nu |_E=\mu$. This shows that there are many doubling measures whose continuous part is doubling.References
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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
- MR Author ID: 214816
- Email: wumin@scut.edu.cn
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 3585-3589
- MSC (2010): Primary 28C15
- DOI: https://doi.org/10.1090/S0002-9939-10-10358-X
- MathSciNet review: 2661557