  The structure of the spectrum of random operators is studied. It is shown
that if the density of states measure of some subsets of the spectrum is zero,
then these subsets are empty. In particular follows that absolute continuity of
the IDS implies singular spectra of ergodic operators is either empty or of
positive measure. Our results apply to Anderson and alloy type models,
perturbed Landau Hamiltonians, almost periodic potentials and models which are
not ergodic.
