Representation of Functions as Absolutely Convergent Fourier-stieltjes Transforms

0. Introduction. Let G be a locally compact Abelian group, and G* its group of (continuous) characters. We write (y, x) to indicate the function on G*XG which is the value of the character y EG* at the point *6G, or, dually, the value of the character xEG at the point y EG*. Borel sets in G are taken to be the smallest cr-algebra of sets containing all closed sets. We consider the set of all regular, countably additive, complex-valued, Borel measures on G. (For measuretheoretic terms not defined here, see [4].) For any such measure || be finite. Such measures are called here bounded Radon measures. The measure ex such that ex(A) = 1 or 0 as A does or does not contain the point xEG is a particularly simple bounded Radon measure. Let 0, there is a compact subset A of G such that |/(*)| <« for xQA. It is known [ô] that a complex linear functional L on Soo(G) which is bounded in the uniform norm has a unique representation as the integral over G with respect to a bounded Radon measure <p :