  A class of translation bounded complex measures, which have the form of
weighted Dirac combs, on locally compact Abelian groups is investigated. Given
such a Dirac comb, we are interested in its diffraction spectrum which emerges
as the Fourier transform of the autocorrelation measure. We present a
sufficient set of conditions to ensure that the diffraction measure is a pure
point measure. Simultaneously, we establish a natural link to the theory of the
cut and project formalism and to the theory of almost periodic measures. Our
conditions are general enough to cover the known theory of model sets, but also
to include examples such as the visible lattice points.
