Decay measures on locally compact abelian topological groups

Abstract

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M^D and M^ND, where M^D is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and M^ND is the set of measures μ ∈ M such that ν ∉ M^D for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μ^D + μ^ND (μ^D ∈ M^D and μ^ND ∈ M^ND) there exist a Borel set A = A(μ) such that μ^D is the restriction of μ to A, therefore the measures μ^D and μ^ND are singular with respect to each other. The measures μ^D and μ^ND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M^D and M^ND.