A stability property for probability measures on Abelian groups

On an arbitrary LCA group G, let a probability measure [mu]2 have the property that it is uniquely defined, up to a shift and a central symmetry, by the modulus of its characteristic function. Then, if [mu]1 is a probability measure on whose characteristic function is an entire function of finite order with real zeros, the property mentioned for [mu]2 remains valid for [mu]=[mu]1x[mu]2 on .