  In this paper we show the existence of the generalized Eberlein decomposition
for Fourier transformable measures with Meyer set support. We prove that each
of the three components is also Fourier transformable and has Meyer set
support. We obtain that each of the pure point, absolutely continuous and
singular continuous components of the Fourier transform is a strong almost
periodic measure, and hence is either trivial or has relatively dense support.
We next prove that the Fourier transform of a measure with Meyer set support is
norm almost periodic, and hence so is each of the pure point, absolutely
continuous and singular continuous components. We show that a measure with
Meyer set support is Fourier transformable if and only if it is a linear
combination of positive definite measures, which can be chosen with Meyer set
support, solving a particular case of an open problem. We complete the paper by
discussing some applications to the diffraction of weighted Dirac combs with
Meyer set support.
