Probabilities on a compact group

Abstract

Let G be an arbitrary compact Hausdorff group. A probability measure on G is a non-negative, real-valued, countably additive, regular Borel measure ,u on G such that ,t(G) = 1. If ,t and X are two probability measures on G then their convolution A * X is also a probability measure on G. In fact if X and Y are any two independent random variables on an arbitrary probability space which take their values in G and if A and X are their respective distributions, then A * X is the distribution of the pointwise product XY. Thus the arithmetic of G-valued independent random variables is just the arithmetic of probability measures on G. This arithmetic is discussed in ?2. Weak* convergence of probability measures is discussed and several theorems concerning infinite powers and infinite products are proved. These and other related problems have been studied in one form or another by other authors. Vorob'yov, in [12], has considered the case in which G is a finite commutative group. Hewitt and Zuckerman, in [3], have studied a class of finite commutative semigroups which includes all finite commutative groups. Kakehashi, in [5], has studied the case in which G is the circle group. In [6], Kawada and Ito have obtained several results for compact metrizable groups. In [14], Wendel has identified all idempotent probability measures when G is an arbitrary compact Hausdorff group. More recently Kloss, in [8], and Urbanik, in [11], have obtained further results in the case that G is an arbitrary compact Hausdorff group. These results and their connection with our findings will be described in the sequel. It is a pleasure to record here the author's indebtedness to Professor Edwin Hewitt for the wealth of valuable advice that he has given during the preparation of this paper. 1. Preliminaries. 1.1. For measure-theoretic terms and techniques not explained explicitly in this paper see [1]. For topological and set-theoretic information see [7]. Reference should be made to [13] for the elementary theory of topological groups and the theory of representations of a compact group. The elementary theory of Banach algebras may be found in [9]. 1.2. Let G be an arbitrary compact Hausdorff group and e(G) the Banach space of all continuous complex-valued functions on G. The class of Borel sets in G, denoted 63, is the smallest o--algebra of subsets of G that contains each open subset of G. A probability measure on G is a non-negative, real-valued,