Note on Fourier-Stieltjes integral. II.
Zirô Takeda · Kodai Mathematical Journal · 1953
lo In the preceding note [5] we intended to prove the following classical theorem concerning Fourier-Stieltjes integral from the stand point of topological group theory* Theorem I (Bochner-Phillips).Let f(x) be a bounded measurable function defined on a locally compact abelian group G satisfying the following condition: (i) for every x^ £ G and complex numbers c^ (/*= l,2,o..n).Then f (x) coincides almost everywhere with the Fourier transform of a bounded Radon measure £ on the dual group δ, i.e. hy hy =5: f G0= α.e.If f(x) is continuous, then the equality holds for all points.