This chapter is devoted to a study of Riesz product measures. For elementary results on Riesz products on the circle see A.I. The class of measures that we shall call “Riesz products” is defined in Section 7.1. The existence and elementary properties of such measures are established also in 7.1. In Section 7.2 we discuss the orthogonality of pairs of Riesz products and show that it is often the case that two Riesz products are either equivalent or mutually singular. In Section 7.3 we give a criterion for a Riesz product to be tame and use that criterion to show that every non-discrete locally compact abelian group supports a tame Hermitian i.p. probability measure in M o (G), a result proved by other methods in Section 6.8 and used in Section 8.2 to show that the Šilov boundary of M(G) is not all ofthe maximal ideal space. Section 7.3 also contains an application to the symbolic calculus for M(G), a topic covered in detail in Chapter 9.
KeywordsProbability Measure Haar Measure Finite Subset Structure Theorem Dual Group
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