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In this paper we show the existence of the generalized Eberlein decompositionfor Fourier transformable measures with Meyer set support. We prove that eachof the three components is also Fourier transformable and has Meyer setsupport. We obtain that each of the pure point, absolutely continuous andsingular continuous components of the Fourier transform is a strong almostperiodic 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 isnorm almost periodic, and hence so is each of the pure point, absolutelycontinuous and singular continuous components. We show that a measure withMeyer set support is Fourier transformable if and only if it is a linearcombination of positive definite measures, which can be chosen with Meyer setsupport, solving a particular case of an open problem. We complete the paper bydiscussing some applications to the diffraction of weighted Dirac combs withMeyer set support.

On the Fourier Analysis of Measures with Meyer Set Support