# Null-finite sets in metric groups and their applications

## TL;DR

This paper introduces null-finite sets in metric groups, explores their properties and connections to Haar-null and Haar-meager sets, and applies these concepts to prove continuity of mid-point convex functions under certain boundedness conditions, resolving a long-standing problem.

## Contribution

It defines null-finite sets and demonstrates their significance in relation to Haar-null and Haar-meager sets, applying these insights to functional equations and inequalities.

## Key findings

- Null-finite Borel sets are Haar-null and Haar-meager.
- Every non-discrete metric Abelian group is the union of two null-finite sets.
- A mid-point convex function is continuous if bounded on a non-null-finite Borel subset.

## Abstract

In the paper we 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 $\mathit{null}$-$\mathit{finite}$ 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.