Representing Measures for Finite-Dimensional Function Spaces

Definitions and mathematical statement

Let $S$ be a topological space, and let $\mathcal B(S)$ denote the sigma-algebra generated by its open sets. Let $C(S,{\mathbb{R}})$ be the real vector space of continuous real-valued functions on $S$. Fix a finite-dimensional real vector subspace $V\subseteq C(S,{\mathbb{R}})$ and a real-linear functional $L\in V^*=\operatorname{Hom}_{{\mathbb{R}}}(V,{\mathbb{R}})$.

The problem asks for necessary and sufficient conditions on $(S,V,L)$ for the existence of a representing Borel measure. Under the usual convention that a measure is nonnegative and countably additive, this means finding conditions equivalent to the existence of a measure \[ \mu:\mathcal B(S)\longrightarrow[0,\infty] \] such that \[ \forall f\in V,\qquad \int_S |f|\,\mathrm d\mu<\infty \quad\text{and}\quad L(f)=\int_S f\,\mathrm d\mu. \] The integrability requirement makes explicit that each displayed integral represents the finite real number $L(f)$.

Unresolved qualifications. The supplied record says only "Borel measure" and does not explicitly specify whether nonnegative or signed measures are intended. The formulation above uses the conventional nonnegative interpretation, which requires confirmation. No assumption that $\mu$ is finite, locally finite, regular, or Radon is supplied. No separation, compactness, or local compactness hypothesis on $S$ is supplied, and $V$ is not assumed to contain the constant function $1$. These additional hypotheses must not be imposed without source clarification. The original wording asks whether a representing measure exists for the given data; the curated wording asks for a characterization of the data for which it exists.

Short English statement

For a finite-dimensional real space of continuous functions on an arbitrary topological space, characterize which linear functionals arise by integration against a Borel measure, with every function in the space integrable. The source does not explicitly settle the sign convention for the measure or impose finiteness or regularity conditions. This statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003444$, title: Representing Measures for Finite-Dimensional Function Spaces; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record supplies proposed year $2017$, but no individual proposer.
• Supplied 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 Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact page or named contributor for this problem is supplied.
• Supplied literature reference: Curto--Fialkow--Moeller, extremal truncated moments. https://arxiv.org/abs/math/0610882.
• Supplied literature reference: Infusino--Kuna--Rota, moment representation. https://doi.org/10.1016/j.jfa.2014.06.012.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports complete criteria in one-dimensional, compact, extremal, and several special algebraic settings, while reporting that no general necessary-and-sufficient characterization for arbitrary finite-dimensional $V$ and $L$ is known. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification; the precise hypotheses of the listed special settings are not supplied.
• The registry labels the statement corrected_verified and explains that notation, typography, and sentence structure were repaired while preserving mathematical content. This is supplied editorial metadata, not an independent mathematical verification of this draft.