Invariance groups and convergence of types of measures on Lie groups

Abstract

Let G be a connected Lie group and let {λi} be a sequence of probability measures on G converging (in the usual weak topology) to a probability measure λ. Suppose that {αi} is a sequence of affine automorphisms of G such that the sequence {αi,(λi)} also converges, say to a probability measure μ. What does this imply about the sequence {αi}? 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, {αi} is relatively compact in the group of all affine automorphisms of ℝn.