The core variety and representing measures in the truncated moment problem

Abstract

The classical Truncated Moment problem asks for necessary and sufficient conditions so that a linear functional $L$ on ${\mathcal{P}}_d$, the vector space of real $n$-variable polynomials of degree at most $d$, can be written as integration with respect to a positive Borel measure $\mu$ on ${\mathbb{R}}^n$. We work in a more general setting, where $L$ is a linear functional acting on a finite dimensional vector space $V$ of Borel-measurable functions defined on a $T_1$ topological space $S$. Using an iterative geometric construction, we associate to $L$ a subset of $S$ called the \textit{core variety}, $\mathcal{C}\mathcal{V}(L)$. Our main result is that $L$ has a representing measure $\mu$ if and only if $\mathcal{C}\mathcal{V}(L)$ is nonempty. In this case, $L$ has a finitely atomic representing measure, and the union of the supports of such measures is precisely $\mathcal{C}\mathcal{V}(L)$. We also use the core variety to describe the facial decomposition of the cone of functionals in the dual space $V^{\ast }$ having representing measures. We prove a generalization of the Truncated Riesz-Haviland Theorem of Curto-Fialkow, which permits us to solve a generalized Truncated Moment Problem in terms of positive extensions of $L$. These results are adapted to derive a Riesz-Haviland Theorem for a generalized Full Moment Problem and to obtain a core variety theorem for the latter problem.