Visible and invisible cantor sets
In this chapter we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff measure-is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set sa...
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2013
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_97808176_v2_n_p11_Cabrelli http://hdl.handle.net/20.500.12110/paper_97808176_v2_n_p11_Cabrelli |
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| Sumario: | In this chapter we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff measure-is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure μ for which the set has positive and finite μ-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e., a Cantor set for which any translation invariant measure is either 0 or non-σ-finite) that enables us to show that the collection of these sets is also dense in the set of all compact subsets of a Polish space X. © Springer Science+Business Media New York 2013. |
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