Positive Radon Measures for Truncated Moment Data

Definitions and mathematical statement

Let $X$ be a locally compact topological space, let $V\subseteq C(X,{\mathbb{R}})$ be a finite-dimensional real vector space with basis $a_1,\ldots,a_m$, and let $s=(s_1,\ldots,s_m)\in{\mathbb{R}}^m$. Here $C(X,{\mathbb{R}})$ denotes the space of continuous real-valued functions on $X$. Define the linear functional $L_s:V\to{\mathbb{R}}$ by \[ L_s\!\left(\sum_{j=1}^m c_j a_j\right)=\sum_{j=1}^m c_j s_j. \] The requested characterization is a necessary and sufficient condition on $(X,V,s)$ for the existence of a positive Radon measure $\mu$ on $X$ such that \[ a_j\in L^1(X,\mu),\qquad s_j=\int_X a_j(x)\,\mathrm d\mu(x) \quad\text{for every }j\in\{1,\ldots,m\}. \] Equivalently, every $f\in V$ must be integrable and satisfy $L_s(f)=\int_X f\,\mathrm d\mu$. Integrability is made explicit because the prescribed moments are finite real numbers; differences of two infinite integrals are not allowed. No assumption of finite total mass, compact support, polynomial functions, or membership of the constant function $1$ in $V$ is supplied.

Unresolved topological qualification. Under a usual locally compact Hausdorff convention, a positive Radon measure is a nonnegative regular Borel measure finite on compact sets. The source assumes only that $X$ is locally compact and does not specify Hausdorffness or its precise convention for Radon measures. The characterization is therefore requested in the source's stated generality, with this convention requiring clarification; Hausdorffness is not added as a hypothesis.

Scope qualification. The source requests a characterization but does not specify an admissible form of criterion or additional hypotheses on $X$, $V$, or $s$. Its supplied open label and literature assessment are curation claims, not independently verified mathematical conclusions.

Short English statement

Characterize which finite lists of prescribed real moments of continuous functions on a locally compact space can be represented by a positive Radon measure, with every prescribed integral finite. The topology and Radon-measure convention need clarification. The supplied open status requires independent review.

Sources

• Local UnsolvedMath problem registry, permanent ID $30003450$, title Positive Radon Measures for Truncated Moment Data, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied original report citation: Real Algebraic Geometry With a View Toward Moment Problems and Optimization (2018), Oberwolfach Reports, pp. 771--862, DOI 10.4171/owr/2017/14. https://doi.org/10.4171/owr/2017/14. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The record separately gives proposed year 2017; the bibliographic year difference is unresolved here.
• Supplied literature reference: Solution of the truncated complex moment problem for flat data. https://doi.org/10.1090/memo/0568.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. It labels the problem open and states that positivity, extension, and flatness criteria exist in important special settings, while the arbitrary positive Radon-measure existence question as extracted is not resolved without further hypotheses on $X$ and the data. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.