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Invariance groups and convergence of types of measures on Lie groups

Published online by Cambridge University Press:  24 October 2008

S. G. Dani
S. G. Dani
Affiliation:
Sonderforschungsbereich-170, Mathematisches Institut, Bunsenstrasse, 3–5, D-3400 Göttingen, Germany
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 91 - 108
Copyright
Copyright © Cambridge Philosophical Society 1992

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