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.
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Notes
- 1.
But if we drop the condition that \(G\) be a Lie group, we should work instead with regular Borel probability measures on the topological group \(G\).
- 2.
If \(G\) is abelian, then the binary operation in the group is usually written additively.
- 3.
It is common in the literature to see the alternative definition “ \(\widehat{\mu }(\pi ) = \int _{G}\pi (g)\mu (dg)\)” which is natural for probabilists but which clashes with the analysts’ convention that we introduced in Chap. 2.
- 4.
Many theorems that we state hold under more general conditions on \(G\). The reader who wants minimal assumptions may consult the original sources, or check what is really needed from the proof.
- 5.
If we were to take a strict analogy with the well-known theory in Euclidean space, we would only use the terminology “standard” Gaussian measure for the case where \(\sigma t = \displaystyle \frac{1}{2}\).
- 6.
i.e. hermitian, if you prefer that terminology.
- 7.
As a result of reading an early version of this manuscript, Herbert Heyer [97] was inspired to prove a new Bochner-type theorem for central probability measures on compact groups.
- 8.
Our definition is slightly different from that of Lo and Ng, who introduce an ordering of the countable set \(\widehat{G}\) and instead of taking arbitrary finite subsets of \(\widehat{G}\) as we do, choose sets of the form \(\{1,2, \ldots , n\}\), with respect to their given ordering.
- 9.
To verify this directly, compute \(\langle f_{\mu }, \pi _{ij}^\prime \rangle \) for each \(\pi ^{\prime } \in \widehat{G}, 1 \le i, j \le d^{\prime }\); it is then a straightforward application of the Peter-Weyl theorem (Corollary 2.2.4) to deduce that \(\widehat{f_{\mu }}(\pi ^{\prime })_{ij} = \widehat{\mu }(\pi ^{\prime })_{ij}\).
- 10.
Our standing hypothesis remains that \(G\) is compact Lie, but observe that the proof of Theorem 4.6.1 requires no use of Lie structure.
- 11.
If \(\rho _{1}, \rho _{2} \in \mathcal{M}(G)\) we write \(\rho _{1} \ge \rho _{2}\) if \(\rho _{1}(f) \ge \rho _{2}(f)\) for all \(f \in C_{c}(G)_{+}\).
- 12.
This term should not be confused with the notion of an amenable group, which is a group that possesses an invariant mean in the sense of the existence of a positive linear functional on \(L^{\infty }(G)\) that is invariant with respect to left, or right translations. Such groups play an important role in ergodic theory, see e.g. Ornstein and Weiss [154].
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© 2014 Springer International Publishing Switzerland
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Applebaum, D. (2014). Probability Measures on Compact Lie Groups. In: Probability on Compact Lie Groups. Probability Theory and Stochastic Modelling, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-319-07842-7_4
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DOI: https://doi.org/10.1007/978-3-319-07842-7_4
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