  In the realm of Delone sets in locally compact, second countable, Hausdorff
groups, we develop a dynamical systems approach in order to study the
continuity behavior of measured quantities arising from point sets. A special
focus is both on the autocorrelation, as well as on the density of states for
random bounded operators. It is shown that for uniquely ergodic limit systems,
the latter measures behave continuously with respect to the Chabauty-Fell
convergence of hulls. In the special situation of Euclidean spaces, our results
complement recent developments in describing spectra as topological limits: we
show that the measured quantities under consideration can be approximated via
periodic analogs.
