The multivariate moment problem is investigated in the general context of the polynomial algebra R[x(i) vertical bar i is an element of Omega] in an arbitrary number of variables x(i), i is an element of Omega. The results obtained are sharpest when the index set Omega is countable. Extensions of Haviland's theorem [17] and Nussbaum's theorem [34] are proved. Lasserre's description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of R[x(i) vertical bar i is an element of Omega] in [27] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author in [30], [32] and [33]. Various results proved in [30], [32] and [33] are shown to continue to hold in this more general setting.
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