  Let $\beta\equiv\beta^{(2n)}$ be an N-dimensional real multi-sequence of
degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv
\mathcal{M}(n)(\beta)$, and let $r:=rank \mathcal{M}(n)$. We prove that if
$\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment
matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique
representing measure \mu, which is r-atomic, with supp \mu$ equal to
$\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$.
Further, \beta has an r-atomic (minimal) representing measure supported in a
semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q}%
\equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ if and only if
$\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving
extension $\mathcal{M}(n+1)$ for which the associated localizing matrices
$\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$ are positive semidefinite
$(1\leq i\leq m)$; in this case, \mu (as above) satisfies supp \mu\subseteq
K_{\mathcal{Q}}$, and \mu has precisely rank \mathcal{M}(n)-rank
\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$ atoms in
$\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.
