Invariant Means on Dedekind complete totally ordered Riesz Spaces

George Chailos, Michael Aristidou

Research output: Contribution to journalArticle

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Abstract

In this paper we consider the set B of all countable bounded subsets of V, where V is a totally ordered σ-Dedekind complete Riesz space equipped with the order topology. We show that on B there exists a function that in a sense behaves as an invariant " mean ". To do this, we construct a set of " approximately invariant means " and we show, using the Ultrafilter Theorem, that this set has a cluster point. This cluster point is the " invariant mean " on B that we are looking for.
Original languageEnglish
Article number3
Pages (from-to)33-47
Number of pages14
JournalTheoretical Mathematics & Applications
Volume6
Issue number3
Publication statusPublished - Aug 2016

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Invariant Mean
Riesz Space
Ordered Space
Order Topology
Ultrafilter
Countable
Subset
Theorem

Keywords

  • Riesz spaces
  • Ultrafilter Theorem
  • Compactness

Cite this

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title = "Invariant Means on Dedekind complete totally ordered Riesz Spaces",
abstract = "In this paper we consider the set B of all countable bounded subsets of V, where V is a totally ordered σ-Dedekind complete Riesz space equipped with the order topology. We show that on B there exists a function that in a sense behaves as an invariant {"} mean {"}. To do this, we construct a set of {"} approximately invariant means {"} and we show, using the Ultrafilter Theorem, that this set has a cluster point. This cluster point is the {"} invariant mean {"} on B that we are looking for.",
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Invariant Means on Dedekind complete totally ordered Riesz Spaces. / Chailos, George; Aristidou, Michael.

In: Theoretical Mathematics & Applications, Vol. 6, No. 3, 3, 08.2016, p. 33-47.

Research output: Contribution to journalArticle

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N2 - In this paper we consider the set B of all countable bounded subsets of V, where V is a totally ordered σ-Dedekind complete Riesz space equipped with the order topology. We show that on B there exists a function that in a sense behaves as an invariant " mean ". To do this, we construct a set of " approximately invariant means " and we show, using the Ultrafilter Theorem, that this set has a cluster point. This cluster point is the " invariant mean " on B that we are looking for.

AB - In this paper we consider the set B of all countable bounded subsets of V, where V is a totally ordered σ-Dedekind complete Riesz space equipped with the order topology. We show that on B there exists a function that in a sense behaves as an invariant " mean ". To do this, we construct a set of " approximately invariant means " and we show, using the Ultrafilter Theorem, that this set has a cluster point. This cluster point is the " invariant mean " on B that we are looking for.

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KW - Ultrafilter Theorem

KW - Compactness

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