  We consider a generic basic semi-algebraic subset $\mathcal{S}$ of the space
of generalized functions, that is a set given by (not necessarily countably
many) polynomial constraints. We derive necessary and sufficient conditions for
an infinite sequence of generalized functions to be realizable on
$\mathcal{S}$, namely to be the moment sequence of a finite measure
concentrated on $\mathcal{S}$. Our approach combines the classical results
about the moment problem on nuclear spaces with the techniques recently
developed to treat the moment problem on basic semi-algebraic sets of
$\mathbb{R}^d$. In this way, we determine realizability conditions that can be
more easily verified than the well-known Haviland type conditions. Our result
completely characterizes the support of the realizing measure in terms of its
moments. As concrete examples of semi-algebraic sets of generalized functions,
we consider the set of all Radon measures and the set of all the measures
having bounded Radon-Nikodym density w.r.t. the Lebesgue measure.
