Probability Measures on Compact Lie Groups
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.