On the determinacy of the moment problem for symmetric algebras of a locally convex space

TL;DR

This paper establishes a sufficient condition for the uniqueness of solutions to the moment problem in symmetric algebras of locally convex spaces, linking measure support properties to determinacy.

Contribution

It provides a new determinacy criterion for the moment problem in symmetric algebras of locally convex spaces, extending previous results.

Findings

A sufficient condition for measure uniqueness based on support containment.

Comparison with existing literature on moment problem determinacy.

Discussion on how prior support knowledge affects measure uniqueness.

Abstract

This note aims to show a uniqueness property for the solution (whenever exists) to the moment problem for the symmetric algebra $S(V)$ of a locally convex space $(V, \tau)$. Let $\mu$ be a measure representing a linear functional $L: S(V)\to\mathbb{R}$. We deduce a sufficient determinacy condition on $L$ provided that the support of $\mu$ is contained in the union of the topological duals of $V$ w.r.t. to countably many of the seminorms in the family inducing $\tau$. We compare this result with some already known in literature for such a general form of the moment problem and further discuss how some prior knowledge on the support of the representing measure influences its determinacy.