On the Support of the Measures in a Symmetric Convolution Semigroup

The aim of the present paper is to prove the following result: Let (#t)t>0 be a vaguely cont inuous convolut ion semigroup consisting of symmetric probabil i ty measures on a locally compact abelian group G. Then there exists a closed subgroup H of G such that supp (#t) = H for all t > 0. As a consequence of this result we get that the support of a symmetric infinitely divisible distribution in 1R" is a closed subgroup of 1RL