We examine the relationship between convergence in the Hausdorff distance and convergence of the associated measures. We introduce a method for estimating the measure of a compact set that is approximated, in the Hausdorff distance sense, by a sequence of compact sets. This is achieved by considering suitable fattenings of the approximating sets and showing that their measures converge.

We then review applications of this result to the study of operator spectra arising from sequences of periodic approximations, leading to the construction of spectral covers.