Null-finite sets in topological groups and their applications

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 (x_n)_n∈ω in X such that for every x ∈ X the set {n ∈ ω : x + x_n ∈ A} 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 B ⊂ G 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.