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We consider a generic basic semi-algebraic subset $\mathcal{S}$ of the spaceof generalized functions, that is a set given by (not necessarily countablymany) polynomial constraints. We derive necessary and sufficient conditions foran infinite sequence of generalized functions to be realizable on$\mathcal{S}$, namely to be the moment sequence of a finite measureconcentrated on $\mathcal{S}$. Our approach combines the classical resultsabout the moment problem on nuclear spaces with the techniques recentlydeveloped to treat the moment problem on basic semi-algebraic sets of$\mathbb{R}^d$. In this way, we determine realizability conditions that can bemore easily verified than the well-known Haviland type conditions. Our resultcompletely characterizes the support of the realizing measure in terms of itsmoments. As concrete examples of semi-algebraic sets of generalized functions,we consider the set of all Radon measures and the set of all the measureshaving bounded Radon-Nikodym density w.r.t. the Lebesgue measure.

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions