Transference in Spaces of Measures

The transference theory for Lp spaces of Calderَn, Coifman, and Weiss is a powerful tool with many applications to singular integrals, ergodic theory, and spectral theory of operators. Transference methods afford a unified approach to many problems in diverse areas, which previously were proved by a variety of methods. The purpose of this paper is to bring about a similar approach to the study of measures. Specifically, deep results in classical harmonic analysis and ergodic theory, due to Bochner, de Leeuw and Glicksberg, Forelli, and others are all extensions of the classical F. & M. Riesz Theorem. We show that all these extensions are obtainable via our new transference principle for spaces of measures.