On the infinite-dimensional moment problem

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with respect to cylinder measures. Our main results provide such integral representations for $A_{+}$–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application we solve the moment problem for the symmetric algebra $S(V)$ of a real vector space $V$. As a byproduct we obtain a new approaches to the moment problem on $S(V)$ for a nuclear space $V$ and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra $A$.