A note on the Kawada–Itô theorem

Abstract

A probability measure μ on a locally compact group G is said to be adapted if the support of μ generates a dense subgroup of G. A classical Kawada–Itô theorem asserts that if μ is an adapted measure on a compact metrizable group G, then the sequence of probability measures 1n∑k=0n−1μkn=1∞ weak∗ converges to the Haar measure on G. In this note, we present a new proof of Kawada–Itô theorem. Also, we show that metrizability condition in the Kawada–Itô theorem can be removed. Some applications are also given.