R-bounded fuzzy measures are equivalent to ε-possibility measures

Abstract

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe nonprobabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which μ(A ∪ B) = max(μ(A), μ(B)). In this paper, we show that possibility measures are, in some sense, universal approximators: for every e > 0, every non-additive measure which satisfies a certain reasonable boundedness property is equivalent to a measure which is e-close to a possibility measure. I. ADDITIVE MEASURES AND FUZZY (NON-ADDITIVE) MEASURES: BRIEF REMINDER Formulation of the problem. One of the main motivations behind fuzzy and other non-probabilistic uncertainty is that the traditional probability theory is sometimes not very adequate for describing uncertainty. From the mathematical viewpoint, probability theory is based on probability (additive) measures. To describe nonprobabilistic uncertainty, researchers therefore came up with a general notion of non-additive (fuzzy) measures, in particular, possibility measures. In this paper, we show that every fuzzy measure satisfying several reasonable properties is isomorphic to an “almost”possibility measure. In order to formulate our result, let us first briefly recall definitions and main properties of probability measures and of fuzzy measures. Definition 1. Let X be a set called a universal set. By an algebra of sets (or algebra, for short) A, we mean a non-empty class of subsets A ⊆ X which is closed under complement and union and intersection, i.e.: • if A ∈ A, then its complement −A also belongs to A; • if A ∈ A and B ∈ A, then A ∪B ∈ A; • if A ∈ A and B ∈ A, then A ∩B ∈ A. Definition 2. By an additive measure, we mean a function μ which maps each set A from some algebra of sets to a nonnegative number μ(A) ≥ 0 and for which μ(A ∪B) = μ(A) + μ(B) for every two sets A and B for which A ∩B = ∅. Comment. Example of additive measures include length of sets on a line, area of planar sets, volume of sets in 3-D space, and probability of different events. Let us recall the main properties of additive measures. The first property is that μ(∅) = 0. Indeed, since the class A is non-empty, it contains some set A. Since A is an algebra, with set A, it also contains sets −A and A ∩ −A = ∅. Since A ∩ ∅ = ∅, additivity implies μ(A) = μ(A) + μ(∅) and thus, μ(∅) = 0. Definition 3. A function μ(A) defined on sets is called monotonic if A ⊆ B implies μ(A) ≤ μ(B). Proposition 1. Every additive measure is monotonic. Proof. Indeed, if A,B ∈ A, then B − A = B ∩ (−A) ∈ A. Due to additivity, we have μ(B) = μ(A) + μ(B − A). Since μ(B − A) ≥ 0, this implies μ(A) ≤ μ(B). The statement is proven. Definition 4. A function μ(A) defined on an algebra of sets is called subadditive if for every two sets A and B, we have μ(A ∪B) ≤ μ(A) + μ(B). Proposition 2. Every additive measure is subadditive. Proof. Indeed, we have B − A ∈ A and, due to additivity, μ(A∪B) = μ(A)+μ(B−A). Since B−A ⊆ B, monotonicity implies that μ(B −A) ≤ μ(B) and thus, μ(A ∪B) ≤ μ(A) + μ(B). The statement is proven. Definition 5. Let X be a set called a universal set. By an σ-algebra A, we mean a non-empty class of subsets A ⊆ X which is closed under complement and countable union and intersection, i.e.: • if A ∈ A, then its complement −A also belongs to A; • if A1 ∈ A, . . . , An ∈ A, . . . , then ∪ i Ai ∈ A; • if A1 ∈ A, . . . , An ∈ A, . . . , then ∩