A Note on the Kawada-Ito Theorem

Abstract

A probability measure mu on a locally compact group G is said to be adapted if the support of mu generates a dense subgroup of G. A classical Kawada-Ito theorem asserts that if mu is an adapted measure on a compact metrizable group G, then the sequence of probability measures {1/n Sigma(n=1)(k=0) mu(k)}(n=1)(infinity) weak* converges to the Haar measure on G. In this note, we present a new proof of Kawada-Ito theorem. Also, we show that metrizability condition in the Kawada-Ito theorem can be removed. Some applications are also given. (C) 2021 Elsevier B.V. All rights reserved.