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_22965009_v_n9780817683788_p11_Cabrelli http://hdl.handle.net/20.500.12110/paper_22965009_v_n9780817683788_p11_Cabrelli |
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paper:paper_22965009_v_n9780817683788_p11_Cabrelli2023-06-08T16:35:27Z Visible and invisible cantor sets Cantor set Cantor space Cantor tree Comeager set Davies set Dimensionless set Generic element Hausdorff measure Polish space Strongly invisible set Tree Visible set 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. 2013 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_22965009_v_n9780817683788_p11_Cabrelli http://hdl.handle.net/20.500.12110/paper_22965009_v_n9780817683788_p11_Cabrelli |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Cantor set Cantor space Cantor tree Comeager set Davies set Dimensionless set Generic element Hausdorff measure Polish space Strongly invisible set Tree Visible set |
| spellingShingle |
Cantor set Cantor space Cantor tree Comeager set Davies set Dimensionless set Generic element Hausdorff measure Polish space Strongly invisible set Tree Visible set Visible and invisible cantor sets |
| topic_facet |
Cantor set Cantor space Cantor tree Comeager set Davies set Dimensionless set Generic element Hausdorff measure Polish space Strongly invisible set Tree Visible set |
| description |
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. |
| title |
Visible and invisible cantor sets |
| title_short |
Visible and invisible cantor sets |
| title_full |
Visible and invisible cantor sets |
| title_fullStr |
Visible and invisible cantor sets |
| title_full_unstemmed |
Visible and invisible cantor sets |
| title_sort |
visible and invisible cantor sets |
| publishDate |
2013 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_22965009_v_n9780817683788_p11_Cabrelli http://hdl.handle.net/20.500.12110/paper_22965009_v_n9780817683788_p11_Cabrelli |
| _version_ |
1768543967667814400 |