Filters and subgroups associated with Hartman measurable functions

Abstract

A bounded function $ϕ: G\to \C$ on an LCA group $G$ is called Hartman measurable if it can be extended to a Riemann integrable function $ϕ^*: X\to \C$ on some group compactification $({\iota }_X,X)$, i.e. on a compact group $X$ such that ${\iota }_X:G\to X$ is a continuous homomorphism with image ${\iota }_X(G)$ dense in $X$ and $\varphi ={\varphi }^{\ast }\circ {\iota }_X$. The concept of Hartman measurability of functions is a generalization of Hartman measurability of sets, which was introduced - with different nomenclature - by S. Hartman to treat number theoretic problems arising in diophantine approximation and equidistribution. We transfer certain results concerning Hartman sets to this more general setting. In particular we assign to each Hartman measurable function $\varphi$ a filter $\F(ϕ)$ on $G$ and a subgroup $\Gamma (\varphi )$ of the dual $\hat{G}$ and show how these objects encode information about the involved group compactification. We present methods how this information can be recovered.

20 pages, 3 figures