ar X iv : m at h / 05 10 06 4 v 1 [ m at h . FA ] 4 O ct 2 00 5 Hartman functions and ( weak ) almost periodicity

In recent papers the concept of Hartman (measurable) sets was investigated. A subset H of the integers, or more generally, of a topological group G is called Hartman measurable (or simply a Hartman set), if H = ι(M) for some continuous homomorphism ι : G → C, C = ι(G) a compact group, and M ⊆ C a set whose topological boundary ∂M is a null set w.r.t. the Haar measure on C. This concept turned out to be useful in the context of coding nα-sequences and, more generally, grouprotations. We investigate the natural generalization from sets to functions via a process similar to the passage from measurable sets to measurable functions. The relation between this class of Hartman (measurable) functions and the class of almost periodic functions is comparable to the relation between Riemann integrable functions and continuous functions. In particular we study the connection of Hartman measurability to (weak) almost periodicity.