  In this paper we investigate the problems 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 we consider the set of all
suitable perturbations called spectrally reasonable measures. In particular, we
exhibit a broad class of spectrally reasonable measures which contains
absolutely continuous ones. On the other hand, we show that except trivial
cases all discrete (purely atomic) measures do not posses this property.
