On continuity of the variation and the Fourier transform

Let S be a commutative semitopological semigroup with identity and involution, F a compact subset in the topology of pointwise convergence of the set of semicharacters on S. Let f be a function which admits a (necessarily unique) integral representation of the form with respect to a complex regular Borel measure/f on F. The function Ifl(’) defined by Ifl(s)-- frp(s)dltzl is called the variation of f. It is shown that the variation Ifl is bounded and continuous if and only if f is also bounded and continuous. This, coupled with the author’s previous characterization of functions of bounded variation, gives a new description of the Fourier transforms of bounded measures on locally compact commutative groups.