Abstract
Many authors have made great strides in extending the celebrated F. and M. Riesz Theorem to various abstract settings. Most notably, we have, in chronological order, the work of Bochner [1], Helson and Lowdenslager [7], de Leeuw and Glicksberg [4], and Forelli [6]. These formidable papers build on each other’s ideas and provide broader extensions of the F. and M. Riesz Theorem. Our goal in this paper is to use the analytic Radon-Nikodým property and prove a representation theorem (Main Lemma 2.2 below) for a certain class of measure-valued mappings on the real line. Applications of this result yield the the main theorems from [4] and [6]. First, we will review briefly the results with which we are concerned, and describe our main theorem. The F. and M. Riesz Theorem states that if a complex Borel measure μ on the circle is such that ∫ π
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