  Suppose that $\sA$ is a finitely generated commutative unital real algebra
and $K$ is a closed subset of the set $\hat{A}$ of characters of $\sA$. We
study the following problem: When is {\it each} linear functional $L:{\sA} \to
\dR$ an integral with respect to some signed Radon measure on $\hat{\sA}$
supported by the set $K$? A complete characterization of the sets $K$ and
algebras $\sA$ by necessary and sufficient conditions is given. The result is
applied to the polynomial algebra $\dR[x_1,\dots,x_d]$ and subsets $K$ of
$\dR^d$.
