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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Every complete doubling metric space
carries a doubling measure


Authors: Jouni Luukkainen and Eero Saksman
Journal: Proc. Amer. Math. Soc. 126 (1998), 531-534
MSC (1991): Primary 28A12; Secondary 54F45
MathSciNet review: 1443161
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Abstract: We prove that a complete metric space $X$ carries a doubling measure if and only if $X$ is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on $X$ and of the homogeneity exponents of $X$ are equal. We also show that a closed subset $X$ of $\mathbf{R}^{n}$ carries a measure of homogeneity exponent $n$. These results are based on the case of compact $X$ due to Vol$^{\prime }$berg and Konyagin.


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Additional Information

Jouni Luukkainen
Affiliation: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland
Email: luukkain@cc.helsinki.fi

Eero Saksman
Affiliation: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland
Email: saksman@cc.helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04201-4
PII: S 0002-9939(98)04201-4
Keywords: Doubling metric space, homogeneous metric space, Assouad dimension, doubling measure, homogeneous measure
Received by editor(s): August 20, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society