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Families of symmetric Cantor sets from the category and measure viewpoints

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Published/Copyright: August 14, 2019
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 26 Issue 4

Abstract

We consider the family 𝒞⁢𝒮 of symmetric Cantor subsets of [0,1]. Each set in 𝒞⁢𝒮 is uniquely determined by a sequence a=(an) belonging to the Polish space X:=(0,1)ℕ equipped with probability product measure μ. This yields a one-to-one correspondence between sets in 𝒞⁢𝒮 and sequences in X. If 𝒜⊂𝒞⁢𝒮, the corresponding subset of X is denoted by 𝒜∗. We study the subfamilies ℋ0, 𝒮⁢𝒫 and ℳ of 𝒞⁢𝒮, consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have ℳ⊂ℋ0⊂𝒮⁢𝒫, and these inclusions are proper. We prove that the sets ℳ∗, ℋ0∗, 𝒮⁢𝒫∗ are residual in X, and μ⁢(ℋ0∗)=0, μ⁢(𝒮⁢𝒫∗)=1.

Keywords: Symmetric Cantor sets; Baire category; product measure; microscopic sets; sets of Hausdorff dimension zero; porous sets
MSC 2010: 28A80; 28A35; 54E52

Dedicated to Professor Alexander Kharazishvili on the occasion of his 70th birthday

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Received: 2018-10-07
Revised: 2018-12-02
Accepted: 2018-12-20
Published Online: 2019-08-14
Published in Print: 2019-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
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https://doi.org/10.1515/gmj-2019-2039
Keywords for this article
Symmetric Cantor sets; Baire category; product measure; microscopic sets; sets of Hausdorff dimension zero; porous sets

Articles in the same Issue

  1. Frontmatter
  2. On absolutely Baire nonmeasurable functions
  3. Existence of special rainbow triangles in weak geometries
  4. Images of Bernstein sets via continuous functions
  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
  10. On the Gitik–Shelah theorem
  11. On equidistant lines of given line configurations
  12. On convergence of sequences of functions possessing closed graphs
  13. On combinatorial and set-theoretical aspects of some finite and infinite point sets
  14. On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
  15. Automorphism groups of mono-unary algebras and CH
  16. On topologies related to the extension of the Lebesgue measure
  17. An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
  18. Local affine selections of convex set-valued functions
  19. On orbits without the Baire property
  20. Mann iteration process for monotone nonexpansive mappings with a graph
  21. Mixing coded systems
  22. Methods of comparison of families of real functions in porosity terms
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