Duality between Measure and Category of Almost All Subsequences of a Given Sequence

Paolo Leonetti; Harry Miller; Leila Miller-Van Wieren

Duality between Measure and Category of Almost All Subsequences of a Given Sequence

Functional Analysis 2017-12-29 v3 General Topology Probability

Authors: Paolo Leonetti , Harry Miller , Leila Miller-Van Wieren

Abstract

Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$ . This provides a non-analogue between measure and category.

Keywords

statistical cluster points subsequences lebesgue measure meager set duality category

Cite

@article{arxiv.1711.04265,
  title  = {Duality between Measure and Category of Almost All Subsequences of a Given Sequence},
  author = {Paolo Leonetti and Harry Miller and Leila Miller-Van Wieren},
  journal= {arXiv preprint arXiv:1711.04265},
  year   = {2017}
}

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5 pages, rewritten

Duality between Measure and Category of Almost All Subsequences of a Given Sequence

Abstract

Keywords

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