  A bounded function $\phi: G\to \C$ on an LCA group $G$ is called Hartman
measurable if it can be extended to a Riemann integrable function $\phi^*: X\to
\C$ 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 $\phi=\phi^*\circ\iota_X$. 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 $\phi$ a filter $\F(\phi)$ on $G$ and a subgroup $\Gamma(\phi)$ 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.
