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Ideal convergent subsequences and rearrangements for divergent sequences of functions

  • Marek Balcerzak EMAIL logo , Michał Popławski and Artur Wachowicz
Published/Copyright: November 30, 2017
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 6

Abstract

Let 𝓙 be an ideal on ℕ which is analytic or coanalytic. Assume that (fn) is a sequence of functions with the Baire property from a Polish space X into a Polish space Z, which is divergent on a comeager set. We investigate the Baire category of 𝓙-convergent subsequences and rearrangements of (fn). Our result generalizes a theorem of Kallman. A similar theorem for subsequences is obtained if (X,μ) is a σ-finite complete measure space and a sequence (fn) of measurable functions from X to Z is 𝓙-divergent μ-almost everywhere. Then the set of subsequences of (fn), 𝓙-divergent μ-almost everywhere, is of full product measure on {0,1}ℕ. Here we assume additionally that 𝓙 has property (G).

MSC 2010: 40A30; 40A05; 54E52; 28A05
Keywords: ideal convergence; Baire category; σ-finite measure; divergence almost everywhere; subsequences; rearrangement

Dedicated to Professor Paolo de Lucia on his 80th birthday

Communicated by Anna De Simone

Acknowledgement

We would like to thank the Referee who has indicated us some important improvements and corrections.

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Received: 2016-4-28
Accepted: 2017-1-6
Published Online: 2017-11-30
Published in Print: 2017-11-27

© 2017 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
https://doi.org/10.1515/ms-2017-0062
Keywords for this article
ideal convergence; Baire category; σ-finite measure; divergence almost everywhere; subsequences; rearrangement

Articles in the same Issue

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
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