Invariance groups and convergence of types of measures on lie groups

Abstract

Let G be a connected Lie group and let {λ _i} be a sequence of probability measures on G converging (in the usual weak topology) to a probability measure λ . Suppose that {a_i} is a sequence of affine automorphisms of G such that the sequence {α _i,(λ _i)} also converges, say to a probability measure μ . What does this imply about the sequence {α _i}? It is a classical observation that if G = Rⁿ for some n, and neither of λ and μ is supported on a proper affine subspace of Rⁿ, then under the above condition, {α _i} is relatively compact in the group of all affine automorphisms of Rⁿ.