Moment problems in an infinite number of variables

Let ^∞ = ×_d=1 ^∞ Given a closed set K ⊆ ^∞ and s: S → where S denotes the set of tuples of nonnegative integers (n₁,n₂,..) with n_d > 0 for finitely many d, the K-moment problem on ^∞ entails determining whether or not there exists a measure σ on ^∞ so that supp σ ⊆ K and s(n) = ∫ ^∞ xⁿ dσ (x) for all n ε S. We prove that σ exists if and only if a natural analogue of the Riesz-Haviland functional 葦_s is K-positive, i.e. if p(x) = σ_{0≤|n|≤ m} p_nxⁿ is any polynomial which is nonnegative for all x ∈ K, then 葦_s(p) = σ _{0≤|n|≤ m} p_ns(n) ≥ 0. We will also provide a sufficient condition for σ to be unique, an analogue of a celebrated theorem of K. Schmüdgen and an application to stochastic processes.