# On subadditive functions upper bounded on a 'large' set

## TL;DR

The paper proves that subadditive functions bounded on large shift-compact sets in certain topological groups are locally bounded everywhere, extending classical results on boundedness and continuity.

## Contribution

It establishes that subadditive functions bounded on shift-compact sets are locally bounded, linking large set properties to function regularity in topological groups.

## Key findings

- Subadditive functions bounded on shift-compact sets are locally bounded.
- Shift-compact sets are significant in the study of functional inequalities.
- Results extend classical boundedness theorems to broader topological group contexts.

## Abstract

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily shift-compact for $X$ completely metrizable (see \cite{BJ} and \cite{BinO8}). Although in general boundedness of a subadditive function does not imply its continuity, here we prove that each subadditive function $f:X\rightarrow \mathbb{R}$ (i.e. with the function satisfying $f(x+y)\leq f(x)+f(y)$ for $x,y\in X$) bounded above on a~shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to \cite[Chapter~XVI]{Kuczma} and papers by N.H.~Bingham and A.J.~Ostaszewski \cite{BO,BinO1,BinO2,BinO6,BinO7}.