  In this paper we characterize the Fourier transformability of a strongly
almost periodic measure in terms of an integrability condition for its Fourier
Bohr series. We also provide a necessary and sufficient condition for a
strongly almost periodic measure to be a Fourier transform of a measure. We
discuss the Fourier transformability of a measure on $\RR^d$ in terms of its
Fourier transform as a tempered distribution. We conclude by looking at a large
class of such measures coming from the cut and project formalism.
