NULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS

In the paper we introduce and study a new family of small sets which is tightly connected with two well known sigma-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 is an element of omega) in X such that for every x is an element of X the set {n is an element of : x + x(n) is an element of 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 Haarnull, 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 subset of 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.