Abstract

Let G be a compact Abelian group with character group X. Let S be a subset of X such that, for some real-valued homomorphism ψ on X, the set S⋂ψ−1(]−∞,ψ(χ)]) is finite for all χ in X. Suppose that μ is a measure in M(G) such that μˆ vanishes off of S, then μ is absolutely continuous with respect to the Haar measure on G.