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Probability Measures on Compact Lie Groups

  • David ApplebaumEmail author
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 70)

Abstract

We introduce the space of Borel probability measures (and the important subspaces of central and symmetric measures) on a group, and topologise this space with the topology of weak convergence. A key tool for studying such measures is the (non-commutative) Fourier transform, which we extend from its action on functions that we described in Chap.  2. We discuss Lo-Ng positivity as a possible replacement for Bochner’s theorem in this context. The theorems of Raikov-Williamson and Raikov are presented that give necessary and sufficient conditions for absolute continuity with respect to Haar measure. We then use the Fourier transform to find conditions for square-integrable densities, and the Sugiura space techniques of Chap.  3 to investigate smoothness of densities. Next we turn our attention to classifying idempotent measures and present the Kawada-Itô equidistribution theorem for the convergence of convolution powers of a measure to the uniform distribution. We introduce and establish key properties of convolution operators, including the notion of associated (sub/super-)harmonic functions. Finally we study some properties of recurrent measures on groups.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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