  We investigate when a linear functional $L$ defined on a linear subspace $B$
of a unital commutative real algebra $A$ admits an integral representation
w.r.t. a positive Radon measure supported on a closed subset $K$ of the
character space of $A$. We provide a criterion for the existence of such a
representation for $L$ when $A$ is equipped with a submultiplicative seminorm.
We then build on this result to prove our main theorem for $A$ not necessarily
equipped with a topology. This allows us to extend well-known classical results
on truncated moment problems.
