Illinois Journal of Mathematics

Upper porous measures on metric spaces

Ville Suomala

Full-text: Open access

Abstract

We show how a standard method of geometric measure theory for providing density estimates may be used in general metric spaces to obtain information on the upper porosity of packing type measures. We also obtain a connection between lower densities and the upper porosity of measures on Euclidean spaces.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 967-980.

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403725

Digital Object Identifier
doi:10.1215/ijm/1254403725

Mathematical Reviews number (MathSciNet)
MR2546018

Zentralblatt MATH identifier
1183.28009

Subjects
Primary: 28A78: Hausdorff and packing measures
Secondary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28A12: Contents, measures, outer measures, capacities

Citation

Suomala, Ville. Upper porous measures on metric spaces. Illinois J. Math. 52 (2008), no. 3, 967--980. doi:10.1215/ijm/1254403725. https://projecteuclid.org/euclid.ijm/1254403725


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References

  • D. Beliaev, E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Smirnov and V. Suomala, Packing dimension of mean porous measures, preprint.
  • D. Beliaev and S. Smirnov, On dimension of porous measures, Math. Ann. 323 (2002), 123–141.
  • A. S. Besicovitch, On linear sets of points of fractional dimension, Math. Ann. 101 (1929), 161–193.
  • A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points II, Math. Ann. 115 (1938), 296–329.
  • C. Cutler, The density theorem and Hausdorff inequality for packing measure in general metric spaces, Illinois J. Math. 39 (1995), 676–694.
  • J.-P. Eckmann, E. Järvenpää and M. Järvenpää, Porosities and dimensions of measures, Nonlinearity 13 (2000), 1–18.
  • K. J. Falconer, Techniques in fractal geometry, Wiley, Chichester, 1997.
  • H. Federer, Geometric measure theory, Springer-Verlag, Berlin, 1969.
  • J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001.
  • E. Järvenpää and M. Järvenpää, Porous measures on $\mathbb{R}^n$: Local structure and dimensional properties, Proc. Amer. Math. Soc. 130 (2002), 419–426.
  • A. Käenmäki and V. Suomala, Nonsymmetric conical upper density and $k$-porosity, to appear in Trans. Amer. Math. Soc.
  • A. Käenmäki and V. Suomala, Conical upper density theorems and porosity of measures, Adv. Math. 217 (2008), 952–956.
  • J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. Lond. Math. Soc. (3) 4 (1954), 257–301.
  • P. Mattila, Distribution of sets and measures along planes, J. Lond. Math. Soc. (2) 38 (1988), 125–132.
  • P. Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge University Press, Cambridge, 1995.
  • M. E. Mera and M. Morán, Attainable values for upper porosities of measures, Real. Anal. Exchange 26 (2000/01), 101–115.
  • M. E. Mera, M. Morán, D. Preiss and L. Zajíček, Porosity, $\sigma$-porosity and measures, Nonlinearity 16 (2003), 247–255.
  • Y. A. Shevchenko, The Vitali covering theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 103 (1989), 11–14.
  • V. Suomala, Local distribution of fractal sets and measures, Ph.D. Thesis, University of Helsinki, 2005; available at http://ethesis.helsinki.fi/julkaisut/mat/matem/vk/suomala/.
  • V. Suomala, On the conical density properties of measures on $\mathbb{R}^n$, Math. Proc. Cambridge Philos. Soc. 138 (2005), 493–512.
  • L. Zajíček, Sets of $\sigma $-porosity and sets of $\sigma $-porosity $(q)$, Časopis Pěst. Mat. 101 (1976), 350–359.
  • L. Zajíček, Porosity and $\sigma$-porosity, Real. Anal. Exchange 13 (1987/88), 314–350.
  • L. Zajíček, On $\sigma$-porous sets in abstract spaces, Abstr. Appl. Anal. (2005), 509–534.