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On uniform properties of doubling measures


Author: Michael Ruzhansky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3413-3416
MSC (2000): Primary 54E35, 54E50, 46A03, 28E15
DOI: https://doi.org/10.1090/S0002-9939-01-05931-7
Published electronically: May 3, 2001
MathSciNet review: 1845020
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove that if $(X,d,\mu )$ is a metric doubling space with segment property, then $\inf r(E)/r(B)>0$ if and only if $\inf \mu (E)/\mu (B)>0$, where the infimum is taken over any collection $\mathcal {C}$ of balls $E, B$ such that $E\subset B\subset X$. As a consequence we show that if $X$ is a linear metric doubling space, then it must be finite dimensional.


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Additional Information

Michael Ruzhansky
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
Address at time of publication: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, England
MR Author ID: 611131
Email: ruzh@maths.ed.ac.uk, ruzh@ic.ac.uk

Received by editor(s): November 23, 1999
Received by editor(s) in revised form: March 24, 2000
Published electronically: May 3, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society