Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11161-X
Published electronically: January 26, 2012
MathSciNet review: 2917099
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.


References [Enhancements On Off] (What's this?)

  • 1. M. Christ.
    A $ T(b)$ theorem with remarks on analytic capacity and the Cauchy integral.
    Colloq. Math., 60/61(2):601-628, 1990. MR 1096400 (92k:42020)
  • 2. M. Csörnyei and V. Suomala.
    Cantor sets and doubling measures.
    Work in progress, 2010.
  • 3. K. J. Falconer.
    Techniques in Fractal Geometry.
    John Wiley & Sons Ltd., England, 1997. MR 1449135 (99f:28013)
  • 4. A.-H. Fan, K.-S. Lau, and H. Rao.
    Relationships between different dimensions of a measure.
    Monatsh. Math., 135(3):191-201, 2002. MR 1897575 (2003g:28014)
  • 5. P. R. Halmos.
    Measure Theory.
    D. Van Nostrand Company, Inc., New York, 1950. MR 0033869 (11:504d)
  • 6. J. Heinonen.
    Lectures on analysis on metric spaces.
    Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • 7. Y. Heurteaux.
    Estimations de la dimension inférieure et de la dimension supérieure des mesures.
    Ann. Inst. H. Poincaré Probab. Statist., 34(3):309-338, 1998. MR 1625871 (99g:28013)
  • 8. T. Hytönen and H. Martikainen.
    Non-homogeneous $ {Tb}$ theorem and random dyadic cubes on metric measure spaces.
    Preprint. arXiv:0911.4387, 2009.
  • 9. E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Rogovin, and V. Suomala.
    Packing dimension and Ahlfors regularity of porous sets in metric spaces.
    Math. Z., 266(1):83-105, 2010. MR 2670673
  • 10. A. Käenmäki, T. Rajala, and V. Suomala.
    Local homogeneity and dimension of measures in doubling metric spaces.
    Preprint. arXiv:1003.2895, 2010.
  • 11. D. G. Larman.
    On Hausdorff measure in finite-dimensional compact metric spaces.
    Proc. London Math. Soc. (3), 17:193-206, 1967. MR 0210874 (35:1759)
  • 12. J. Luukkainen and E. Saksman.
    Every complete doubling metric space carries a doubling measure.
    Proc. Amer. Math. Soc., 126(2):531-534, 1998. MR 1443161 (99c:28009)
  • 13. S.-M. Ngai.
    A dimension result arising from the $ L\sp q$-spectrum of a measure.
    Proc. Amer. Math. Soc., 125(10):2943-2951, 1997. MR 1402878 (97m:28007)
  • 14. C. A. Rogers.
    Hausdorff measures.
    Cambridge University Press, London, 1970. MR 0281862 (43:7576)
  • 15. E. Saksman.
    Remarks on the nonexistence of doubling measures.
    Ann. Acad. Sci. Fenn. Math., 24(1):155-163, 1999. MR 1678044 (2000b:28006)
  • 16. P. Tukia. Hausdorff dimension and quasisymmetric mappings, Math. Scand., 65(1):152-160, 1989. MR 1051832 (92b:30026)
  • 17. A. L. Vol $ ^{\prime }$berg and S. V. Konyagin.
    A homogeneous measure exists on any compactum in $ {\bf R}^n$.
    Dokl. Akad. Nauk SSSR, 278(4):783-786, 1984. MR 765294 (86d:28018)
  • 18. A. L. Vol $ ^{\prime }$berg and S. V. Konyagin.
    On measures with the doubling condition.
    Izv. Akad. Nauk SSSR Ser. Mat., 51(3):666-675, 1987. MR 903629 (88i:28006)
  • 19. J.-M. Wu.
    Hausdorff dimension and doubling measures on metric spaces.
    Proc. Amer. Math. Soc., 126(5):1453-1459, 1998. MR 1443418 (99h:28016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28C15, 54E50

Retrieve articles in all journals with MSC (2010): 28C15, 54E50


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: https://doi.org/10.1090/S0002-9939-2012-11161-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

American Mathematical Society