  Given a cut and project scheme and a pre-compact Borel window we show that
almost surely all positions of the window give rise to point sets with
Besicovitch almost periodic Dirac combs. In particular, all those positions
lead to pure point diffractive point sets with autocorrelation given by the
covariogram of the window and diffraction given by the square of the Fourier
transform of the window. Moreover the Fourier--Bohr coefficients exist and the
intensity of the Bragg peaks is given by their absolute value square. We show
the existence of an ergodic measure with pure point dynamical spectrum such
that all those window positions give rise to point sets which are generic for
this measure. We complete the paper by reanalysing the class of weak model sets
of maximal density.
