On the factor sets of measures and local tightness of convolution semigroups over lie groups

Abstract

It is shown that for a large class of Lie groups (called weakly algebraic groups) including all connected semisimple Lie groups the following holds: for any probability measure μ on the Lie group the set of all two-sided convolution factors is compact if and only if the centralizer of the support of μ in G is compact. This is applied to prove that for any connected Lie groupG, any homomorphism of any real directed (submonogeneous) semigroup into the topological semigroup of all probability measures onG is locally tight.