Let β≡β(2n) be an N-dimensional real multi-sequence of
degree 2n, with associated moment matrix M(n)≡M(n)(β), and let r:=rankM(n). We prove that if
M(n) is positive semidefinite and admits a rank-preserving moment
matrix extension M(n+1), then M(n+1) has a unique
representing measure \mu, which is r-atomic, with supp \muequalto\mathcal{V}(\mathcal{M}(n+1)),thealgebraicvarietyof\mathcal{M}(n+1).Further,βhasanr−atomic(minimal)representingmeasuresupportedinasemi−algebraicsetK_{\mathcal{Q}}subordinatetoafamily\mathcal{Q}%
\equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]ifandonlyif\mathcal{M}(n)ispositivesemidefiniteandadmitsarank−preservingextension\mathcal{M}(n+1)forwhichtheassociatedlocalizingmatrices\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])arepositivesemidefinite(1\leq i\leq m);inthiscase,μ(asabove)satisfiessuppμ⊆KQ​, 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$.Comment: 33 pages; to appear in J. Operator Theor