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. © Springer Science+Business Media New York 2013.
Registro:
| Documento: |
Parte de libro
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| Título: | Visible and invisible cantor sets |
| Autor: | Cabrelli, C.; Darji, U.B.; Molter, U. |
| Filiación: | Departamento de Matemática FCEyN, Universidad de Buenos Aires C1428EGA C.A.B.A., Argentina IMAS - CONICET, Buenos Aires, Argentina
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States
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| Palabras clave: | Cantor set; Cantor space; Cantor tree; Comeager set; Davies set; Dimensionless set; Generic element; Hausdorff measure; Polish space; Strongly invisible set; Tree; Visible set; Forestry; Fractals; Cantor sets; Cantor spaces; Cantor tree; Comeager set; Davies set; Dimensionless set; Generic element; Hausdorff measures; Strongly invisible set; Tree; Visible set; Topology |
| Año: | 2013
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| Volumen: | 2
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| Página de inicio: | 11
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| Página de fin: | 21
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| DOI: |
http://dx.doi.org/10.1007/978-0-8176-8379-5_2 |
| Título revista: | Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center
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| Título revista abreviado: | Excursions in Harmon. Anal.: The Febr. Fourier Talks at the Norbert Wien. Cent.
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| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_97808176_v2_n_p11_Cabrelli |
Referencias:
- Besicovitch, A.S., Taylor, S.J., On the complementary intervals of a linear closed set of zero Lebesgue measure (1954) J. London Math. Soc, 29, pp. 449-459
- Best, E., A closed dimensionless linear set (1939) Proc. Edinburgh Math. Soc, 2 (6), pp. 105-108
- Cabrelli, C., Hare, K.E., Molter, U.M., Classifying Cantor sets by their fractal dimensions (2010) Proc. Amer. Math. Soc, 138 (11), pp. 3965-3974
- Cabrelli, C., Mendivil, F., Molter, U.M., Shonkwiler, R., On the h-Hausdorff measure of Cantor sets (2004) Pac. J. Math, 217, pp. 29-43
- Davies, R.O., Sets which are null or non-sigma-finite for every translation-invariant measure (1971) Mathematika, 18, pp. 161-162
- Elekes, M., Keleti, T., Borel sets which are null or non-cr -finite for every translation invariant measure (2006) Adv. Math, 201 (1), pp. 102-115
- Kechris, A.S., (1995) Descriptive Set Theory, Graduate Texts in Mathematics, 156. , Springer-Verlag New York
- Rogers, C.A., (1998) Hausdorff Measures, Cambridge Math Library, , Cambridge University Press, Cambridge
Citas:
---------- APA ----------
Cabrelli, C., Darji, U.B. & Molter, U.
(2013)
. Visible and invisible cantor sets. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center, 2, 11-21.
http://dx.doi.org/10.1007/978-0-8176-8379-5_2---------- CHICAGO ----------
Cabrelli, C., Darji, U.B., Molter, U.
"Visible and invisible cantor sets"
. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center 2
(2013) : 11-21.
http://dx.doi.org/10.1007/978-0-8176-8379-5_2---------- MLA ----------
Cabrelli, C., Darji, U.B., Molter, U.
"Visible and invisible cantor sets"
. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center, vol. 2, 2013, pp. 11-21.
http://dx.doi.org/10.1007/978-0-8176-8379-5_2---------- VANCOUVER ----------
Cabrelli, C., Darji, U.B., Molter, U. Visible and invisible cantor sets. Excursions in Harmon. Anal.: The Febr. Fourier Talks at the Norbert Wien. Cent. 2013;2:11-21.
http://dx.doi.org/10.1007/978-0-8176-8379-5_2