Truncated K-moment problems in several variables

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}$,$1\leq i\leq m$.