Convergence Theorems for Generalized Random Variables and Martingales

We examine generalizations of random variables and martingales. We prove a new convergence theorem for setvalued martingales. We also generalize a well known characterization of set-valued random variables to the fuzzy setting. Keywords— Banach space, fuzzy, martingale, random variable, set-valued. 1 Set-valued random variables and martingales The theory of conditional expectations and martingales has been established for Banach space-valued, Bochnerintegrable functions. Hiai and Umegaki have then generalized random variables, conditional expectations and martingales to the set-valued (multivalued) setting in [1]. We will assume that the reader is familiar with the basic ideas of random variables and provide a brief introduction to the set-valued generalization and also to the related concept of conditional expectations which has played an important role in probability theory, ergodic theory and quantum statistical mechanics. We also contribute to the theory of setvalued martingales and since martingales are important in probability theory, we feel that there are potential applications to be developed from this work. This paper can be divided into two main results. The first result is an original convergence theorem for set-valued martingales. The second result is a generalization of a useful characterization ofmeasurable set-valued random variables by Hiai and Umegaki in the fuzzy setting. The original theorem by Hiai and Umegaki was an essential tool used to obtain many of the subsequent results in [1] and it is clear that we can apply this new generalized theorem in an analagous way in the fuzzy setting. We will be again considering a measure space (Ω,Σ, μ), a separable Banach space (X, ‖·‖)withK a field of scalars (either R or C). We remind the reader that functions f, g from Ω intoX are said to be equal almost everywhere, denoted f = g a.e. (μ) or f(ω) = g(ω) (∀ω ∈ Ω) a.e. (μ) if f(ω) = g(ω) for all ω ∈ Ω except on a set of measure zero. We can omit reference to μ and Ω if there is no confusion. That is f = g a.e. if μ({ω ∈ Ω : f(ω) = g(ω)}) = 0. We will denote by M [Ω, X] and L[Ω, X] the collections of measurable and p-integrable functions f : Ω → X respectively, for 1 ≤ p ≤ ∞. Due to the completeness of X we have several characterizations of this class of functions (see [1] Theorem 1.0). As in [1], K(X) shall denote the collection of nonempty compact subsets of X and Kk(X) the collection of convex sets in K(X). We denote the collection of (weakly) measurable set-valued functions F : Ω → Kk(X) by M[Ω, Kk(X)]. That is F ∈ M[Ω, Kk(X)] if and only if for all O open, O ⊂ X, F(O) ∈ Σ. Let p be chosen such that 1 ≤ p ≤ ∞ then we denote by L[Ω, Kk(X)] = L[Ω,Σ, μ,Kk(X)] the space of all p-integrable functions in M[Ω, Kk(X)], where two functions F1, F2 ∈ L[Ω, Kk(X)] will be considered identical if F1(ω) = F2(ω) a.e. The topology on Kk(X) is induced by the Hausdorff metric defined by dH(A,B) = max { sup a∈A inf b∈B ‖a− b‖, sup b∈B inf a∈A ‖a− b‖ } , (1.1) for A,B ⊆ X. It is well known (Kk(X), dH) is a complete separable metric space. It now follows that in this topology (Fn)n∈N ⊆ M[Ω, Kk(X)] then Fn −→ F as n −→ ∞ if and only if dH(Fn(ω), F (ω)) −→ 0 for all ω ∈ Ω a.e. We can consider (X, ‖·‖) to be a topological space since a norm defines a natural metric on X which induces the metric topology on X. We define the closure of a set A ⊆ X, denoted cl(A) to be the norm closure of A. That is, the smallest closed set containingA with respect to the metric topology induced by the norm ‖ · ‖. A measurable function f : Ω → X is called a measurable selection of F if f(ω) ∈ F (ω) for all ω ∈ Ω a.e. That is f is a measurable selection of F if f(ω) ∈ F (ω) for all ω ∈ Ω except on a set of measure zero. We define the set S F = {f ∈ L [Ω, X] : f(ω) ∈ F (ω) a.e.}. It is easy to show that S p F is a closed subset of L [Ω, X] for 1 ≤ p ≤ ∞. Lemma 1.1 [1] Let F ∈ M[Ω, Kk(X)] and 1 ≤ p ≤ ∞. If S F is nonempty, then there exists a sequence (fi)i∈N contained in S p F such that F (ω) = cl{fi(ω)} for ω ∈ Ω. Corollary 1.2 [1] Let F1, F2 ∈ M[Ω, Kk(X)] and 1 ≤ p ≤ ∞. If S F1 = S p F2 = ∅ then F1(ω) = F2(ω), for all ω ∈ Ω. A brief discussion of the generalization of conditional expectations is necessary before we reach our first main result. This material is covered more comprehensively in [1]. Let W be a sub-σ-algebra of Σ, S F (W ) = {f ∈ L [Ω,W, μ,X] : f(ω) ∈ F (ω)a.e.}. (1.2) We define ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009