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 → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f : G → C is a Hartman function if there exists a group compactification (ι, C) and F : C → C such that f = F • ι and F is Riemann integrable, i.e. the set of discontinuities of F is a null set w.r.t. the Haar measure. In particular we determine how large a compactification for a given group G and a Hartman function f : G → C must be, to admit a Riemann integrable representation of f . The connection to (weakly) almost periodic functions is investigated. 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.

On splittings and integration of almost periodic functions with and without geometry

“…The next corollary is different to [35,Thm. 4.12], as it provides assumptions on range to obtain an almost periodic reversible part, whereby Ruess/Summers provide conditions on the Eberlein weakly almost periodic f , to obtain an integral in this class.…”

“…A generalization of Kadets' integration theorem for general groups has been investigated by Basit [5]. Applications of Kadets' theorem have been studied by Ruess and Summers [34,35] and more recently by Farkas and Kreidler [16], just to mention a few.…”