Kreps-yan Theorem for Banach Ideal Spaces

Let C be a closed convex cone in a Banach ideal space X on a measurable space with a σ-finite measure. We prove that conditions C ∩ X+ = {0} and C ⊃ −X+ imply the existence of a strictly positive continuous functional on X , whose restriction to C is non-positive. Let (Ω,F ) be a measurable space, which is complete with respect to a measure (that is, a countably-additive function) μ : F 7→ [0,∞]. Consider the vector space L(μ) = L(Ω,F , μ) of classes of μ-equivalent a.s. finite F -measurable functions. This space is a vector lattice (Riesz space) with respect to the natural order structure, induced by the cone L0+(μ) of non-negative elements [1], [2]. Let X be a solid subspace (ideal) in L(μ), that is, X is a linear subset of L(μ) and conditions x ∈ X, |y| ≤ |x| imply that y ∈ X. Assume that there is a norm on X, satisfying the condition ‖x‖ ≤ ‖y‖, if |x| ≤ |y|, x, y ∈ X (monotone norm) and X is complete with respect to this norm. As is known, in this case (X, ‖ · ‖) is called a Banach ideal space on (Ω,F , μ) [1], [3]. Let X be a Banach ideal space with the non-negative cone X+ = {x ∈ X : x ≥ 0}. An element g of the topological dual space X ′ is called strictly positive if 〈x, g〉 := g(x) > 0, x ∈ X+\{0}. Slightly modifying the terminology of [4], we say that X has the Kreps-Yan property, if for any closed convex cone C ⊂ X, satisfying the conditions C ∩X+ = {0}, −X+ ⊂ C, (1) there exists a strictly positive element g ∈ X ′ such that 〈x, g〉 ≤ 0, x ∈ C. This property can be considered also in a more general setting of a locally convex space with a cone. We refer to [4] for the results in this direction as well as for further references and comments, concerning the papers of Kreps [5] and Yan [6]. It is said that a topological space (X, τ) has the Lindelöf property, if every open cover of X has a countable subcover. If the weak topology σ(X,X ) of a Banach space X has the Lindelöf property (for brevity, X is weakly Lindelöf) and the set of strictly positive functionals g ∈ X ′ is non-empty, then X has the Kreps-Yan property [7, Theorem 1.1]. A space X is weakly Lindelöf if any of the following conditions hold true: (a) X is reflexive, (b) X is separable, (c) X is weakly compactly generated (see [8]). Also, it is known that the space L(Ω,F ,P) of F -measurable essentially bounded functions (where P is a probability measure) has the Kreps-Yan property. This was established in [7, Theorem 2.1] (and, independently, in [9]). Note 2000 Mathematics Subject Classification. 46E30, 46B42.