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Null-finite sets in topological groups and their applications

  • Taras Banakh
  • Eliza Jabłónska
Article
  • 17 Downloads

Abstract

In the paper we introduce and study a new family of “small” sets which is tightly connected with two well known σ-ideals: of Haar-null sets and of Haar-meager sets. We define a subset A of a topological group X to be null-finite if there exists a convergent sequence (xn)nω in X such that for every xX the set {nω : x + xnA} is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function f : G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a subset B which is not null-finite and whose closure is contained in G. This gives an alternative short proof of a known generalization of the Bernstein–Doetsch theorem (saying that a mid-point convex function f: G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a non-empty open subset B of G). Since Borel Haar-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function f: G → ℝ defined on an open convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borel subset BG which is not Haar-null or not Haar-meager in X. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Deparetment of MathematicsIvan Franko National University of LvivLvivUkraine
  2. 2.Institute of MathematicsJan Kochanowski University in KielceKielcePoland
  3. 3.Institute of MathematicsPedagogical University of CracowKrakówPoland

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