Asymptotic behaviour of measures under automorphisms

Abstract

Various problems concerning probability measures on locally compact groups involve understanding what happens in the limit when a measure, or a sequence of measures, is operated upon by a sequence of automorphisms of the group. In the case of Lie groups it turns out that much of the thrust of the questions can be reduced to studying the behaviour of measures on Euclidean spaces under linear transformations. Our aim here is to describe some simple properties in this respect and their applications to various problems; convergence of types, concentration functions, factor compactness, Levy's measures are some of the topics to which applications will be made.