  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 $A$.
