Visible and invisible cantor sets

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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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Detalles Bibliográficos
Autores principales:	Cabrelli, C., Darji, U.B., Molter, U.
Formato:	CHAP
Materias:	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 Hausdorff measures Topology
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_97808176_v2_n_p11_Cabrelli
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_97808176_v2_n_p11_Cabrelli
record_format	dspace
spelling	todo:paper_97808176_v2_n_p11_Cabrelli2023-10-03T16:42:52Z Visible and invisible cantor sets Cabrelli, C. Darji, U.B. Molter, U. 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 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. CHAP info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_97808176_v2_n_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 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
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 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 Cabrelli, C. Darji, U.B. Molter, U. 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 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
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.
format	CHAP
author	Cabrelli, C. Darji, U.B. Molter, U.
author_facet	Cabrelli, C. Darji, U.B. Molter, U.
author_sort	Cabrelli, C.
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
url	http://hdl.handle.net/20.500.12110/paper_97808176_v2_n_p11_Cabrelli
work_keys_str_mv	AT cabrellic visibleandinvisiblecantorsets AT darjiub visibleandinvisiblecantorsets AT molteru visibleandinvisiblecantorsets
_version_	1807324026159235072

Visible and invisible cantor sets

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