The strong conclusion of the F. and M. Riesz theorem on groups

Abstract

Let S S be a closed proper generating subsemigroup of the dual Γ \Gamma of a locally compact abelian group G G . Then there are Haar singular measures on G G orthogonal to S S unless G = R × Δ G = {\mathbf {R}} \times \Delta or T × Δ {\mathbf {T}} \times \Delta with Δ \Delta discrete, and then all μ \mu orthogonal to S S are Haar absolutely continuous.