Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 797-804

GENERALIZED HAUSDORFF MEASURE FOR GENERIC COMPACT SETS

Richárd Balka and András Máthé

Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
P.O. Box 127, 1364 Budapest, Hungary; balka.richard 'at' renyi.mta.hu

University of Warwick, Mathematics Institute
Coventry, CV4 7AL, United Kingdom; A.Mathe 'at' warwick.ac.uk

Abstract. Let X be a Polish space. We prove that the generic compact set K \subseteq X (in the sense of Baire category) is either finite or there is a continuous gauge function h such that 0 < H^h(K) < \infty, where H^h denotes the h-Hausdorff measure. This answers a question of Cabrelli, Darji, and Molter. Moreover, for every weak contraction f : K -> X we have H^h(K \cap f(K)) = 0. This is a measure theoretic analogue of a result of Elekes.

2010 Mathematics Subject Classification: Primary 28A78.

Key words: Contraction, dimension function, exact Hausdorff dimension, gauge, generic compact set, Polish space, typical.

Reference to this article: R. Balka and A. Máthé: Generalized Hausdorff measure for generic compact sets. Ann. Acad. Sci. Fenn. Math. 38 (2013), 797-804.

Full document as PDF file

doi:10.5186/aasfm.2013.3835

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