Integral representations in conuclear spaces

Abstract

The main result can be formulated as follows: Let Γ be a closed convex cone in a quasi-complete conuclear space whose dual contains a countable total subset (e.g. R ^N, D′, E′, S′, any nuclear Frechet space). Assume that Γ satisfies condition (C) : For every compact convex set K⊂Γ, the set of points ‘between’ the origin and points of K (i.e. Γ ∩ K − Γ) is compact. Then every point of Γ has an integral representation by means of extreme generators of Γ; this integral representation is unique if and only if Γ is a lattice.

More precisely: let S be a subset of Γ, not containing the origin, having one point on each ray, and such that UλS is a Borel set (such sets can be constructed). Let S _e be the set of points of S lying on extreme rays of Γ. Let M⁺ (S _e) be the set of Radon measures on S _e, having finite first moments, such that the weak integrals ∝xdm(x) belong to Γ. Then A) for every point a ∈ Γ there exists m ∈M⁺ (S_e) such that a=∝xdm(x). B) for every a ∈ Γ there is precisely one m ∈ M⁺(S _e) such that a=∝xdm(x), if and only if Γ is a lattice.

The condition (C) implies that Γ is proper (Γ∩−Γ=(o)). Any weakly complete proper convex cone in a quasi-complete conuclear space satisfies condition (C). Other examples are given.

In the last part of the paper we propose an abstract generalization: ‘conuclear’ cones. The cones in a conuclear space satisfying condition (C), and the cones with compact base in an arbitrary space are both conuclear.

Method. In order to prove these results we work, not with Radon measures on the rather arbitrary sets S defined above, but with the notion of conical measure defined by G. Choquet. We restrict ourselves, however, to those conical measures which can be defined by means of integrals: localizable conical measures.