Artículo
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: | Artículo |
| Título: | Visible and invisible cantor sets |
| Autor: | Cabrelli, C.; Darji, U.B.; Molter, U. |
| Filiación: | Departamento de Matemática FCEyN, Universidad de Buenos Aires, IMAS - CONICET, C.A.B.A., Argentina Department of Mathematics, University of Louisville, Louisville, KY, United States |
| 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 |
| Año: | 2013 |
| Número: | 9780817683788 |
| Página de inicio: | 11 |
| Página de fin: | 21 |
| DOI: | http://dx.doi.org/10.1007/978-0-8176-8379-5_2 |
| Título revista: | Applied and Numerical Harmonic Analysis |
| Título revista abreviado: | Appl. Numer. Harmon. Anal. |
| ISSN: | 22965009 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_22965009_v_n9780817683788_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-∏-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, Vol. 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. Applied and Numerical Harmonic Analysis(9780817683788), 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" . Applied and Numerical Harmonic Analysis, no. 9780817683788 (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" . Applied and Numerical Harmonic Analysis, no. 9780817683788, 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. Appl. Numer. Harmon. Anal. 2013(9780817683788):11-21.http://dx.doi.org/10.1007/978-0-8176-8379-5_2
