Maldistributed sequences in metric spaces

Let a kind of measure μ be given on the set of all positive integers N. By a μ-maldistributed sequence in a metric space (X,d) we mean any sequence possessing μ-extremely many points in each nonempty open subspace of (X,d). Let s(X) denote the metric space of all sequences in (X,d) endowed with the...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2025-01, Vol.541 (2), p.128667, Article 128667
Main Authors: Mišík, Ladislav, Tóth, János T.
Format: Article
Language:English
Subjects:
Maldistributed sequence
Measure on sets of positive integers
Nowhere dense set
Residual set
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Summary:Let a kind of measure μ be given on the set of all positive integers N. By a μ-maldistributed sequence in a metric space (X,d) we mean any sequence possessing μ-extremely many points in each nonempty open subspace of (X,d). Let s(X) denote the metric space of all sequences in (X,d) endowed with the Fréchet metric. In this paper we study topological properties of the set of all maldistributed sequences in X as a subspace s(X). We show that if (X,d) is separable then this set is residual for a large class of commonly used measures μ on N.
ISSN:0022-247X
DOI:10.1016/j.jmaa.2024.128667