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 $$\mathcal{M}$(n)\equiv $\mathcal{M}$(n)(\beta)$,andlet$r:=rank $\mathcal{M}$(n)$\mathrm{.}Weprovethatif$$\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$equalto$$\mathcal{V}$($\mathcal{M}$(n+1))$,thealgebraicvarietyof$$\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 $$\mathcal{Q}$% \equiv{q_{i}}{i=1}^{m}\subseteq$\mathbb{R}$[t{1},…,t_{N}]$ifandonlyif$$\mathcal{M}$(n)$ is positive semidefinite and admits a rank-preserving extension $\mathcal{M}(n+1)$ for which the associated localizing matrices ${\mathcal{M}}_{q_i}(n+[\frac{1+\mathrm{deg}{q}_i}{2}])$ are positive semidefinite $(1\le i\le m)$; 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}])$atomsin$$\mathcal{Z}$(q_{i})\equiv {t\in$\mathbb{R}$^{N}:q_{i}(t)=0}$,$1\leq i\leq m$.