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 μ ˆ vanishes off of S , then μ is absolutely continuous with respect to the Haar measure on G .