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

 

Existence of doubling measures via generalised nested cubes


Authors: Antti Käenmäki, Tapio Rajala and Ville Suomala
Journal: Proc. Amer. Math. Soc. 140 (2012), 3275-3281
MSC (2010): Primary 28C15; Secondary 54E50
Published electronically: January 26, 2012
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Abstract: Working on doubling metric spaces, we construct generalised
dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each $ \varepsilon >0$ there is a doubling measure having full measure on a set of packing dimension at most $ \varepsilon $.


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Additional Information

Antti Käenmäki
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
Email: antti.kaenmaki@jyu.fi

Tapio Rajala
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, I56127 Pisa, Italy
Email: tapio.rajala@sns.it

Ville Suomala
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
Address at time of publication: Department of Mathematics and Statistics, P.O. Box 3000, FI-90014 University of Oulu, Finland
Email: ville.suomala@jyu.fi, ville.suomala@oulu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11161-X
PII: S 0002-9939(2012)11161-X
Keywords: Doubling measure, nested cubes in metric spaces
Received by editor(s): November 19, 2010
Received by editor(s) in revised form: March 30, 2011
Published electronically: January 26, 2012
Additional Notes: The third author acknowledges the support of the Academy of Finland, project #126976
Communicated by: Tatiana Toro
Article copyright: © Copyright 2012 American Mathematical Society