  We consider the class of all linear functionals $L$ on a unital commutative
real algebra $A$ that can be represented as an integral w.r.t. to a Radon
measure with compact support in the character space of $A$. Exploiting a recent
generalization of the classical Nussbaum theorem, we establish a new
characterization of this class of moment functionals solely in terms of a
growth condition intrinsic to the given linear functional. To the best of our
knowledge, our result is the first to exactly identify the compact support of
the representing Radon measure. We also describe the compact support in terms
of the largest Archimedean quadratic module on which $L$ is non-negative and in
terms of the smallest submultiplicative seminorm w.r.t. which $L$ is
continuous. Moreover, we derive a formula for computing the measure of each
singleton in the compact support, which in turn gives a necessary and
sufficient condition for the support to be a finite set. Finally, some aspects
related to our growth condition for topological algebras are also investigated.
