Null-finite sets in metric groups and their applications

We shall introduce 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 an infinite compact subset $K\subset X$ such that for every $x\in X$ the intersection $K\cap (x+A)$ 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 $f:G\to \mathbb R$ 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 Bernstein-Doetsch theorem (saying that a mid-point convex function $f:G\to \mathbb R$ defined on an open covex 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 null-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function $f:G\to \mathbb R$ 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 $B\subset G$ 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. Details can be found in the joint paper (https://arxiv.org/abs/1706.08155) with Eliza Jablonska.