First Results for a Mathematical Theory of Possibilistic Processes

This paper provides the measure theoretic basis for a theory of possibilistic processes. We generalize the definition of a product τ field to an indexed family of τ -fields, without imposing an ordering on the index set. We also introduce the notion ‘measurable cylinder’ and show that any product τ -field can be generated by its associated field of measurable cylinders. Furthermore, we introduce and study the notions ‘τ -subspace’, ‘extension of a τ -space’ and ‘one-point extension of a τ -space’. Using these notions, we prove that for any family of possibility distributions (πT ′ | ∅ ⊂ T ′ b T ), satisfying a natural consistency condition, a family (ft | t ∈ T ) of possibilistic variables can be constructed such that the possibilistic variable ×t∈T ′ft (with ∅ ⊂ T ′ b T ) has πT ′ as a possibility distribution. As a special case we obtain a possibilistic analogon of the probabilistic Daniell-Kolmogorov theorem, a cornerstone for the theory of stochastic processes. 1 Preliminary notions In this paper, we develop the mathematical and topological apparatus necessary for proving a possibilistic analogon for the well-known theorem of DaniellKolmogorov [Doob, 1967]. This theorem is the cornerstone for the mathematical theory of stochastic processes. In short, it tells us that, given a family of realvalued functions on finite Cartesian powers of a sample space that satisfy natural consistency conditions, there exists a basic space, a probability measure on that basic space, and a family of stochastic variables that have these real-valued functions as their probability distribution functions. The results in this paper are the possibilistic counterparts. ∗Postdoctoral Fellow of the Belgian National Fund for Scientific Research (NFWO). In this section, we give a number of basic definitions, which are needed in the following sections. A subset L of the power class P(X) of a nonempty set X is called a plump field [Wang and Klir, 1992] on X iff it is closed under arbitrary unions and intersections. The atom of L containing the element x ∈ X is defined as [ x ]L def = ⋂ {A | A ∈ L and x ∈ A}. Furthermore, a subset A of X is an atom of L iff (∃x ∈ X)(A = [ x ]L). The set of the atoms of L is denoted by XL. It is readily verified that XL ⊆ L and that (∀x ∈ X)(x ∈ [x ]L). Finally, for any subset A of X, A ∈ L ⇔ A = ⋃ x∈A [ x ]L. A τ -field or ample field [Wang, 1982] R on a nonempty set X is a plump field on X that is closed under complementation. The couple (X,R) is called a τ -space. Finally [Wang and Klir, 1992], for a nonempty set X and a subset A of its power class P(X), τX(A) denotes the smallest τ -field on X which includes A. Furthermore, if A is a subset of a nonempty set X, then we write A b X iff A is a finite subset of X. Throughout this paper, X denotes a nonempty set, R is a τ -field on X, (L,≤) denotes a complete lattice with greatest element 1 and smallest element 0, and 1L is the identical permutation of L. 2 Possibilistic Processes De Cooman and Kerre [1993] have generalized Zadeh’s original definition of a possibility measure as follows: if (L,≤) is a complete lattice and (X,R) is a τ -space, then aR−L mapping Π is a (L,≤)-possibility measure on (X,R) iff for any family (Aj | j ∈ J) of elements of R