Decomposition of analytic measures on groups and measure spaces

This paper provides a new approach to proving generalizations of the F.&M. Riesz Theorem, for example, the result of Helson and Lowdenslager, the result of Forelli (and de Leeuw and Glicksberg), and more recent results of Yamagushi. We study actions of a locally compact abelian group with ordered dual onto a space of measures, and consider those measures that are analytic, that is, the spectrum of the action on the measure is contained within the positive elements of the dual of the group. The classical results tell us that the singular and absolutely continuous parts of the measure (with respect to a suitable measure) are also analytic. The approach taken in this paper is to adopt the transference principle developed by the authors and Saeki in another paper, and apply it to martingale inequalities of Burkholder and Garling. In this way, we obtain a decomposition of the measures, and obtain the above mentioned results as corollaries.