Non-compact versions of Edwards’ Theorem

Abstract

Edwards' Theorem establishes duality between a convex cone in the space of continuous functions on a compact space and the set of representing or Jensen measures for this cone. In this paper we prove non-compact versions of this theorem.

On Caratheodory Completeness in C^n

“…We have the following result (which follows from a version of Hahn-Banach theorem, see e.g., [6]). Theorem 4.…”

“…In 2013 Gogus, Perkins, and Poletsky [6] proved the following non-compact version of Edwards' theorem Theorem 8. Let X be a locally compact σ-compact Hausdorff space and let S ⊂ C(X) be a convex cone.…”

“…This part of the paper is motivated by [6] and we use methods of that paper. Let X be a topological space and let C(X) (resp.…”

“…Let X be a normal topological space. We say that X is of GP P -type if for any positive linear functional Λ : C(X) → R there exists a compact set K ⊂ X such that Λ(ϕ) = 0 for any ϕ ∈ C(X) with ϕ = 0 on K. According to [6] any locally compact σ-compact Hausdorff space is of GP P -type. We have another example.…”

Carathéodory completeness on the plane

“…The implication (2) =⇒ (1) in any dimension is claimed in [2]. However, the proof is based on a false version of Hahn-Banach theorem, claimed in [5]. So, we give a new proof on the complex plane.…”