not have a singular component. The present note contributes further evidence in this direction by showing that in many groups (including the real line and the integers) there are relatively small sets on which the Fourier transform of any absolutely continuous measure can be so modified that it becomes the transform of a singular measure. Let r be the dual of a locally compact abelian group G; L1(G) and M(G) denote the spaces of all Haar-integrable functions on G and of all complex Borel measures on G, respectively, and we identify L'(G) with the absolutely continuous members of M(G). The Fourier transform of ,u E M(G) is defined to be