Porous measures on : Local structure and dimensional properties

Authors:
Esa Järvenpää and Maarit Järvenpää

Journal:
Proc. Amer. Math. Soc. **130** (2002), 419-426

MSC (2000):
Primary 28A12, 28A80

DOI:
https://doi.org/10.1090/S0002-9939-01-06161-5

Published electronically:
June 8, 2001

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We study dimensional properties of porous measures on . As a corollary of a theorem describing the local structure of nearly uniformly porous measures we prove that the packing dimension of any Radon measure on has an upper bound depending on porosity. This upper bound tends to as porosity tends to its maximum value.

**[BS]**D. B. Beliaev and S. K. Smirnov,*On dimension of porous measures*, preprint.**[EJJ]**J.-P. Eckmann, E. Järvenpää, and M. Järvenpää,*Porosities and dimensions of measures*, Nonlinearity**13**(2000), 1-18. CMP**2000:07****[Fa]**K. J. Falconer,*Techniques in Fractal Geometry*, John Wiley & Sons, Chichester, 1997. MR**99f:28013****[Fe]**W. Feller,*An Introduction to Probability Theory and Its Applications*, John Wiley & Sons, New York, 1950. MR**12:424a****[JJ]**E. Järvenpää and M. Järvenpää,*Porous measures on the real line have packing dimension close to zero*, preprint (http://www.math.jyu.fi/research/papers.html, number 212).**[JJM]**E. Järvenpää, M. Järvenpää, and R. Daniel Mauldin,*Deterministic and random aspects of porosities*, preprint (http://www.math.jyu.fi/research/papers.html, number 221).**[M1]**P. Mattila,*Distribution of sets and measures along planes*, J. London Math. Soc. (2)**38**(1988), 125-132. MR**89f:28019****[M2]**P. Mattila,*Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability*, Cambridge University Press, Cambridge, 1995. MR**96h:28006****[MM]**M. E. Mera and M. Morán,*A zero-one half law for porosity of measures*, preprint.**[S]**A. Salli,*On the Minkowski dimension of strongly porous fractal sets in*, Proc. London Math. Soc. (3)**62**(1991), 353-372. MR**91m:28008**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
28A12,
28A80

Retrieve articles in all journals with MSC (2000): 28A12, 28A80

Additional Information

**Esa Järvenpää**

Affiliation:
Department of Mathematics, P.O. Box 35, University of Jyväskylä, FIN-40351 Jyväskylä, Finland

Email:
esaj@math.jyu.fi

**Maarit Järvenpää**

Affiliation:
Department of Mathematics, P.O. Box 35, University of Jyväskylä, FIN-40351 Jyväskylä, Finland

Email:
amj@math.jyu.fi

DOI:
https://doi.org/10.1090/S0002-9939-01-06161-5

Received by editor(s):
June 13, 2000

Published electronically:
June 8, 2001

Communicated by:
David Preiss

Article copyright:
© Copyright 2001
American Mathematical Society