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].