On subadditive functions upper bounded on a 'large' set

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 $f:X\to \mathbb{R}$ (i.e. with the function satisfying $f(x+y)\le f(x)+f(y)$ 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}.