Compactifications, Hartman functions and (weak) almost periodicity

Abstract

In this paper we investigate Hartman functions on a topological group $G$. Recall that $(ι, C)$ is a group compactification of $G$ if $C$ is a compact group, $ι: G\to C$ is a continuous group homomorphism and $ι(G)$ is dense in $C$. A complex-valued bounded function $f$ on $G$ is a Hartman function if there exists a group compactification $(ι, C)$ and a complex-valued bounded function $F$ on $C$ such that $f=F\circι$ and $F$ is Riemann integrable, i.e. the set of discontinuities of $F$ is a null set with respect to the Haar measure. In particular we answer the question how large a compactification for a given group $G$ and a Hartman function $f$ must be, to admit a Riemann integrable representation of $f$. In order to give a systematic presentation which is self-contained to a reasonable extent, we include several separate sections on the underlying concepts such as finitely additive measures on Boolean set algebras, means on algebras of functions, integration on compact spaces, compactifications of groups and semigroups, the Riemann integral on abstract spaces, invariance of measures and means, continuous extensions of transformations and operations to compactifications, etc.