Moment problem in infinitely many variables

The multivariate moment problem is investigated in the general context of the polynomial algebra $\mathbb{R}[x_i \mid i \in Ω]$ in an arbitrary number of variables $x_i$, $i\in Ω$. The results obtained are sharpest when the index set $Ω$ is countable. Extensions of Haviland's theorem [Amer. J. Math., 58 (1936) 164-168] and Nussbaum's theorem [Ark. Math., 6 (1965) 179-191] 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 $\mathbb{R}[x_i \mid i \in Ω]$ in [Trans. Amer. Math. Soc., 365 (2013) 2489-2504] 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.

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