The Levy continuity theorem for nuclear groups

Let G be an abelian topological group. The Levy continuity theorem says tha t if G is an LCA group, then it has the following property (PL): a sequence of Radon probability measures on G is weakly convergent to a Radon probabi lity measure mu if and only if the corresponding sequence of Fourier transf orms is pointwise convergent to the Fourier transform of mu. Boulicaut [Bo] proved that every nuclear locally convex space G has the property (PL). In this paper we prove that the property (PL) is inherited by nuclear groups, a variety of abelian topological groups containing LCA groups and nuclear locally convex spaces, introduced in [B1].