Mathematische Annalen

, Volume 331, Issue 3, pp 557–576 | Cite as

Average homogeneity and dimensions of measures

  • Esa JärvenpääEmail author
  • Maarit Järvenpää


We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.


Porosity Uniform Distribution Hausdorff Dimension General Notion Dimension Effect 
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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläFinland

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