  The multivariate moment problem is investigated in the general context of the
polynomial algebra $\mathbb{R}[x_i \mid i \in \Omega]$ in an arbitrary number
of variables $x_i$, $i\in \Omega$. The results obtained are sharpest when the
index set $\Omega$ 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
$\mathbb{R}[x_i \mid i \in \Omega]$ 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.
