  A theorem of Siebert asserts that if a sequence of semigroups of probability
measures on a Lie group G is weakly convergent to a semigroup of the same type,
then the corresponding generating functionals are convergent in the weak
operator topology in every unitary representation of the group.The aim of this
note is to give a simple proof of the theorem and propose some improvements. In
particular, we completely avoid employing unitary representations by showing
simply that under the same hypothesis the generating functionals are convergent
pointwise on C^2(G). As a corollary,the above thesis of Siebert is extended to
strongly continuous representations of G on Banach spaces.
