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Journal of the London Mathematical Society Advance Access originally published online on August 14, 2009
Journal of the London Mathematical Society 2009 80(2):514-530; doi:10.1112/jlms/jdp040
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© 2009 London Mathematical Society

Packing dimension of mean porous measures

D. Beliaev

Department of Mathematics
Princeton University
Fine Hall
Washington Road
Princeton, NJ 08544-1000
USA
dbeliaev@math.princeton.edu

E. Järvenpää

Department of Mathematical Sciences
Pentti Kaiteran katu 1
PO Box 3000
90014 University of Oulu
Finland

M. Järvenpää

Department of Mathematical Sciences
Pentti Kaiteran katu 1
PO Box 3000
90014 University of Oulu
Finland
Maarit.Jarvenpaa@oulu.fi

A. Käenmäki

Department of Mathematics and Statistics
PO Box 35 (MaD)
40014 University of Jyväskylä
Finland
antakae@maths.jyu.fi

T. Rajala

Department of Mathematics and Statistics
PO Box 35 (MaD)
40014 University of Jyväskylä
Finland
tamaraja@maths.jyu.fi

S. Smirnov

Department of Mathematics
University of Geneva
2-4 rue du Lièvre, Case postale 64
1211 Genève 4
Switzerland
Stanislav.Smirnov@math.unige.ch

V. Suomala

Department of Mathematics and Statistics
PO Box 35 (MaD)
40014 University of Jyväskylä
Finland
visuomal@maths.jyu.fi

We prove that the packing dimension of any mean porous Radon measure on Rd may be estimated from above by a function which depends on mean porosity. The upper bound tends to d – 1 as mean porosity tends to its maximum value. This result was stated in D. B. BELIAEV and S. K. SMIRNOV [‘On dimension of porous measures’, Math. Ann. 323 (2002) 123–141], and in a weaker form in E. JÄRVENPÄÄ and M. JÄRVENPÄÄ [‘Porous measures on Rn: local structure and dimensional properties’, Proc. Amer. Math. Soc. (2) 130 (2002) 419–426], but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure µ on R such that µ(A) = 0 for all mean porous sets A sub R.

2000 Mathematics Subject Classification 28A75, 28A80.

E.J., M.J., A.K., T.R. and V.S. acknowledge the support of the Academy of Finland (projects # 211229 and # 114821) and the Centre of Excellence in Analysis and Dynamics Research. D.B., E.J. and S.S. acknowledge the support of the Swiss National Science Foundation. T.R. appreciates the financial support of Vilho, Yrjö and Kalle Väisälä Foundation.

Received July 6, 2007; revised May 27, 2009; published online August 14, 2009.


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