The core variety and representing measures in the truncated moment problem

TL;DR

This paper introduces the core variety concept in the truncated moment problem, providing necessary and sufficient conditions for the existence of representing measures and describing their structure.

Contribution

It generalizes the classical truncated moment problem by defining the core variety and linking it to the existence and structure of representing measures.

Findings

Core variety is nonempty iff a representing measure exists.

Representing measures are finitely atomic with supports equal to the core variety.

Generalizes the Riesz-Haviland Theorem for broader moment problems.

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}, $\mathcal{CV}(L)$. Our main result is that $L$ has a representing measure $\mu$ if and only if $\mathcal{CV}(L)$ is nonempty. In this case, $L$ has a finitely atomic representing measure, and the union of the supports of such measures is precisely $\mathcal{CV}(L)$. We also use the core variety to describe the facial decomposition of the cone of functionals in the dual space $V^{*}$ 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.