Continuous measures on compact Lie groups

Abstract

We study continuous measures on a compact semisimple Lie group G using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on G which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if C is a compact set of continuous measures on G there exists a singular measure ν such that ν*μ is absolutely continuous with respect to the Haar measure on G for every μ in C. In Section 4 we show that if f is a finite linear combination of characters then there exist two singular measures μ and ν on G such that f=μ*ν. In the final section we obtain a Wiener-type characterization of a continuous measure on a symmetric space of compact type G/K.