Spectrally reasonable measures

The problems under study are related to measures with a natural spectrum (equal to the closure of the set of the values of the Fourier–Stieltjes transform). Since it is known that the set of all such measures does not have a Banach algebra structure, the set of all suitable perturbations called spectrally reasonable measures is considered. In particular, a broad class of spectrally reasonable measures is exhibited, which contains absolutely continuous ones. On the other hand, it is shown that except trivial cases all discrete (purely atomic) measures do not posses this property.