An autocorrelation and a discrete spectrum for dynamical systems on metric spaces

We study dynamical systems. (X, G, m) with a compact metric space X, a locally compact, sigma-compact, abelian group G and an invariant Borel probability measure m on X. We show that such a system has a discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays, for general dynamical systems, a similar role to that of the autocorrelation measure in the study of aperiodic order for special dynamical systems based on point sets.