The core variety and representing measures in the truncated moment problem

Abstract

The truncated moment problem asks for conditions so that a linear functional L on 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 μ on Rn. More generally, let L act on a finite dimensional space of Borel-measurable functions defined on a T1 topological space S. Using an iterative geometric construction, we associate to L a subset of S called the \textit{core variety} CV(L). Our main result is that L has a representing measure μ if and only if 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 CV(L).