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Excursions in Harmonic Analysis, Volume 2 pp 11–21Cite as

Birkhäuser

Visible and Invisible Cantor Sets

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Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

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.

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References

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Acknowledgments

C. Cabrelli and U. Molter are partially supported by Grants UBACyT X638 and X502 (UBA) and PIP 112-200801-00398 (CONICET). U. Darji is partially supported by University of Louisville Project Initiation Grant.

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Authors and Affiliations

  1. Departamento de Matemática FCEyN, Universidad de Buenos Aires C1428EGA C.A.B.A., Argentina IMAS - CONICET, Buenos Aires, Argentina

    Carlos Cabrelli & Ursula Molter

  2. Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA

    Udayan B. Darji

Authors
  1. Carlos Cabrelli
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  2. Udayan B. Darji
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  3. Ursula Molter
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Corresponding author

Correspondence to Ursula Molter .

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Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

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Cabrelli, C., Darji, U.B., Molter, U. (2013). Visible and Invisible Cantor Sets. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8379-5_2

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