First Results for a Mathematical Theory of Possibilistic Processes

First Results for a Mathematical heory of Possibilistic Processes H.J. Janssen, G. de Cooman and E.E. Kerre Universiteit Gent Vakgroep oegepaste Wiskunde en Informatica Krijgslaan 281, B-9000 Gent, Belgium { Hugo.Janssen, Gert.deCooman, Etienne.Kerre}@rug.ac.be Abstract his 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 (π ), satisfying a natural consistency condition, a family (f t t ) of possibilistic variables can be constructed such that the possibilistic variable f t (with ) has π 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 Daniell- Kolmogorov [Doob, 1967]. his 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. he 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. he atom of L containing the element def x X is defined as [ x ] L = {A A L and x A}. Furthermore, a subset A of X is an atom of L iff ( x X)(A = [ x ] L ). he set of the atoms of L is denoted by X L. It is readily verified that X L 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. he 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), τ X (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 X iff A is a finite subset of X. hroughout 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 1 L 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 a R L mapping Π is a (L, )-possibility measure on (X, R) iff for any family (A j j J) of elements of R Π( A j ) = sup Π(A j ). (1) j J j J he triple (X, R, Π) is called a (L, )-possibility space. Furthermore, if A is a subset of the power class P(X) of X and Π is any A L mapping, then a X L mapping π is called a distribution of Π iff for any A in

A: Π(A) = sup π(x). (2) x A In particular, De Cooman has proven that every (L, )-possibility measure Π on a τ-space (X, R) possesses a unique distribution π that is constant on the atoms of R, and is given by ( x X)(π(x) = Π([ x ] R )). (3) Finally, if A 1 P(X 1 ) and A 2 P(X 2 ), where X 1 and X 2 are nonempty sets, then a X 1 X 2 mapping f is called A 1 A 2 measurable iff ( B A 2 )(f 1 (B) A 1 ). In particular [De Cooman, 1993], if (Ω, R Ω, Π Ω ) is a (L, )-possibility space and (X, R) is a τ-space, then a Ω X mapping f is a possibilistic variable in (X, R) iff f is a R Ω R measurable mapping. he τ-space (X, R) in which the possibilistic variable takes its values, is called the sample space of f, and (Ω, R Ω, Π Ω ) is called the basic space of f. hroughout this paper, we shall assume that any possibilistic variable has the (L, )-possibility space (Ω, R Ω, Π Ω ) as its basic space. Finally, if π Ω represents the distribution of Π Ω and f is a possibilistic variable in (X, R), then the X L mapping π f, which is given for any x X by π f (x) = sup w f 1 ([ x ] R ) π Ω (ω), is called the possibility distribution of f. Using the previous notions, we define a family of possibilistic variables, which is more general than a possibilistic process. Definition 1 (Family of possibilistic variables) Let be a nonempty set. A family (f t t ), such that ( t )(f t is a possibilistic variable in a τ-space (X t, R t )), is called a family of possibilistic variables in the family ((X t, R t ) t ) of τ-spaces, with index set. A possibilistic process will be defined as a family of possibilistic variables, such that the possibilistic variables of this family have the same sample space. his leads to the following definition. Definition 2 (Possibilistic process) Let be a nonempty set. A family (f t t ), such that ( t )(f t is a possibilistic variable in a τ-space (X, R)), is a possibilistic process in the τ-space (X, R), with index set. If the index set is countable, then (f t t ) is called a discrete possibilistic process. A process (f t t ), for which the index set is a real interval, is called a continuous possibilistic process. 3 Product τ-spaces First, we introduce the notions Cartesian product of a family of sets, projection mapping from a Cartesian product and product mapping. Definition 3 Let (X t t ) be a family of nonempty sets with nonempty index set. hen, the Cartesian product of (X t t ) is the set X t of all X t mappings x, such that ( t )(x(t) X t ). In particular, if ( t )(A t X t ), then A t is the subset of X t, which contains all the elements x of X t, such that ( t )(x(t) A t ). If ( t )(X t = X), then the Cartesian product X t is also denoted by X. For any s, pr,s is the X t X s mapping, defined by ( x X t )(pr,s (x) = x(s)), and is called the s-th projection mapping from X t onto X s. Finally, let A be any set. hen, for any family (f t t ) of A X t mappings f t, the unique mapping f : A X t, such that ( t )(f t = pr,t f), is denoted by f t and is called the product mapping of (f t t ). Using the definitions above we can define the notion ychonov topology or product topology, which is a special case of a weak topology induced on a set (see [Kelley, 1959; Willard, 1970]). Definition 4 Let X be any set and ((X t, t ) t ) a family of topological spaces with nonempty index set. he ychonov topology or product topology on X t is the weak topology induced on X t by the family (pr,t t ) of projections, and is denoted by W((X t, t ) t ). hroughout this section, is a nonempty set and ((X t, R t ) t ) is a family of τ-spaces with index set. Furthermore, the closure operator τ X t is abbreviated by τ. Let (X 1, R 1 ) and (X 2, R 2 ) be τ-spaces. Wang [1982] has defined the product of R 1 and R 2 as the τ-field on X 1 X 2 R 1 R 2 def = τ({a 1 A 2 A 1 R 1 and A 2 R 2 }) (4) where τ = τ X1 X 2, and he has proven that for any (x 1, x 2 ) X 1 X 2 [ (x 1, x 2 ) ] R1 R 2 = [ x 1 ] R1 [ x 2 ] R2. (5) he following definition generalizes Wang s original definition towards the product of an indexed family of τ-fields, without imposing an ordering on the index set. Definition 5 R t denotes the smallest τ-field on X t, such that for any s pr,s is a R t R s measurable mapping. R t is called the product τ-field on X t of the family (R t t ) of τ-fields and

( X t, R t) is called the product τ-space of the family ((X t, R t ) t ) of τ-spaces. In case ( t )(R t = R), where R is a τ-field on X, R t is also denoted by R. he following theorem generalizes (4) and (5). heorem 6 he product τ-field R t satisfies: R t = τ({ A t ( t )(A t R t )}) (6) he atoms of the product τ-field R t are characterized by [ x ] where x X t. R t = [ x(t) ] Rt = pr 1,t ([ x(t) ] R t ), (7) he following definition introduces the notion of measurable cylinder of a product τ-space. Definition 7 For any set such that, let pr, be the mapping from X t onto X t, such that ( x X t )(pr, (x) = x ), where x is the restriction of the mapping x to the domain. hen, let C, = {pr 1, (E) E R t }. Any element of C, is called a measurable - cylinder of ( X t, R t). Furthermore, let C = C,. hen any element of C is a measurable cylinder of ( X t, R t). he following properties can be proven. Proposition 8 1. C is a field on X t. 2. C is a base for the product topology W((X t, R t ) t ) on X t. 3. τ(c ) = τ(w((x t, R t ) t )) = R t. 4. he following statements are equivalent. (a) C is a τ-field on X t. (b) C is a plump field on X t. (c) here exists a set such that and C = C,. (d) C = R t. 5. Moreover, if is a finite set, then C is a τ-field on X t. 4 τ-subspaces and extensions of τ-spaces We first define the notions τ-subspace and extension of a τ-space. Definition 9 Let (X, R X ) and (Y, R Y ) be τ-spaces. hen (X, R X ) is called a τ-subspace of (Y, R Y ), and we write (X, R X ) (Y, R Y ) iff R X = {E X E R Y }. (8) It follows immediately from (8) that X Y. Furthermore, it is easily verified that {E X E R Y } is a τ-field on X, so that definition 9 is meaningful. We now introduce the notion of an extension of a τ- space, which generalizes the notion of coarseness [De Cooman and Kerre, 1993]. Definition 10 Let (X, R X ) and (Y, R Y ) be τ-spaces. hen (Y, R Y ) is called an extension of (X, R X ), and we write (X, R X ) (Y, R Y ) iff R X R Y. (9) Condition (9) implies that X Y, and if X = Y, then R X is coarser than R Y. he following property can easily be proven. Proposition 11 is a partial order on the set of τ- subspaces of any τ-space and is a partial order on the set of extensions of any τ-space. Definition 9 gives rise to the notion extension of a (L, )-possibility space as follows. Definition 12 A (L, )-possibility space (Y, R Y, Π Y ) is called an extension of the (L, )- possibility space (X, R X, Π X ) iff (X, R X ) (Y, R Y ) and Π Y RX = Π X. Since any τ-space (X, R) is a topological space, the the following results can be shown to hold. Proposition 13 Let (X, R) be a τ-space. hen (X, R) is a compact topological space iff X R is finite. A subset A of X is closed and compact in (X, R) iff A is a finite union of atoms of R. By the previous proposition, a τ-space (X, R) is not always a compact topological space, However [Kelley, 1959], a noncompact topological space (X, R) can always be embedded in a compact topological space. In particular, if we let be the set R {X \ G G is a closed, compact set in (X, R)} (10) where X = X { } and X, then (X, ) is a compact topological space, and is called a one-point compactification of (X, R). If (X, R) is a compact topological space, then let X = X and = R. Furthermore, if R = τ X ( ), then, according to definition 10, (X, R ) is an extension of (X, R). his leads to the following definition.

Definition 14 Let (X, R) be a τ-space. hen (X, R ) is called a *-extension of (X, R). (X, R ) is called a one-point extension of (X, R) iff (X, R) is a noncompact topological space. For a one-point extension (X, R ) of (X, R), (X, ) is called the one-point compactification associated with (X, R ). Using the previous definition, we obtain the following proposition. Proposition 15 Let (X, R) be a τ-space and let (X, R ) be a *-extension of (X, R). hen (X, R) (X, R ) and (X, R) (X, R ). In particular, 1. if (X, R) is compact, then R = = R and XR = X R, 2. if (X, R ) is a one-point extension of (X, R), then R = τ X (R), XR = X R {{ }}, and R R. 5 Main result In this section, let ((X t, R t ) t ) be a family of τ-spaces with nonempty index set and let s. We can associate a *-extension (Xs, Rs) with the τ-space (X s, R s ). According to definition 14, if (Xs, R s) is a one-point extension of (X s, R s ), then Xs = X s { s }, where s X s and using proposition 15, R s = τ X s (s ) = τ X s (R s ), in which (Xs, s ) is the one-point compactification associated with (Xs, R s). In case (X s, R s ) is a compact topological space, (Xs, R s) coincides with (X s, R s ). Furthermore, if pr,s is the s-th projection mapping from Xt onto Xs, then R t is the smallest τ-field on Xt, such that ( s )(pr,s is a R t R s measurable mapping). Also, if is a nonempty, finite subset of, pr, denotes the mapping from Xt onto Xt, such that ( x Xt )(pr, (x) = x ). For any nonempty, finite subset of, we define C, 1 = {pr, (E) E R t }. hen C = C, is, in accordance with definition 7 and proposition 8, the field of all measurable cylinders of ( Xt, R t ). Finally, let O = {pr 1, (O) O W((X t, t ) t )}, (11) C = {pr 1, (E) E R t }. (12) he following properties can be proven. Proposition 16 1. C O C 2. R t = τ Xt (C ) = τ X t (W((X t, t ) t )) = τ X t (O ). 3. For any set such that, it follows that ( X t, R t ) ( X t, R t ) and ( X t, R t ) ( X t, R t ). We are now ready to proceed to the main topic of this paper, namely, finding a possibilistic counterpart for the probabilistic Daniell-Kolmogorov theorem. We want to prove that, if we have a family of L-valued functions on finite Cartesian powers of a sample space, and if these functions satisfy natural consistency conditions, then we can always find a basic space with possibility measure, and a family of possibilistic variables that have these L-valued functions as their possibility distribution functions. As a matter of fact, we prove a more general result, of which the possibilistic Daniell-Kolmogorov theorem turns out to be a special case. Let (π ) be a family of distributions such that π is the distribution of a (L, )-possibility measure Π on the τ-space ( X t, R t ) for any nonempty, finite subset of. For such family of distributions, we introduce the following consistency condition. Definition 17 (π ) is called consistent iff for any two sets 1 and 2 such that 1 2, we have: ( ) ( x 1 X t ) π 1 (x) = sup π 2 (y) pr 2, 1 (y)=x. (13) Furthermore, for any set such that, let π denote the X t L mapping, given for any x Xt by { π π (x) = (x) if x X t. (14) 0 otherwise If Π denotes the (L, )-possibility measure on ( Xt, R t ) with distribution π, then Π Rt= Π and using proposition 16.3, it fol- lows that ( Xt, R t, Π ) is an extension of the (L, )-possibility space ( X t, R t, Π ). With the above notations, we obtain directly the following result. Proposition 18 he family (π ) is consistent iff the family (π ) is consistent. Recently, Boyen et al. [1995] have generalized Wang s definition [Wang, 1985; Wang and Klir, 1992] of P-consistency for set mappings as follows.

Definition 19 Let X be a nonempty set, A a family of subsets of X and (L, ) a complete lattice. A A L mapping Π is called P-consistent iff for any family (A j j J) of elements of A and any element A of A: A j J A j Π(A) sup j J Π(A j ). he notion of P-consistency was introduced by Wang [Wang, 1985; Wang and Klir, 1992] in the context of the extension of set mappings with ([0, 1], ) as codomain to ([0, 1], )-possibility measures. he notion of extendability is generalized by Boyen et al. [1995] as follows. Definition 20 Let X be a nonempty set, A a family of subsets of X and (L, ) a complete lattice. A A L mapping Π is extendable to a (L, )-possibility measure on a τ-space (X, R) iff there exists a (L, )- possibility measure Π on (X, R) such that ( A A)(Π(A) = Π (A)). Π is called extendable to a (L, )- possibility measure iff there exists an ample field R on X such that Π is extendable to a (L, )-possibility measure on (X, R). he following theorem [Boyen et al., 1995] is needed to prove the theorems that follow. heorem 21 Let X be a nonempty set, A be a family of subsets of X and (L, ) a complete lattice. hen, for any P-consistent A L mapping Π, any of the following conditions is sufficient for the extendability of Π. (E 1 ) (L, ) is a complete chain. (E 2 ) A is a plump field. (E 3 ) (L, ) = (B, ), where B is a plump field on some set Y. Moreover, the complete lattice (L, ) can always be embedded using a supremum preserving mapping φ into a second complete lattice (L, ), in such a way that for any P-consistent A L mapping Π, φ Π is a P-consistent A L mapping which is extendable to a (L, )-possibility measure. Now, we are ready to construct a possibility measure on the product τ-space ( Xt, R t ) by means of the family (π ). herefore, we need the following result. heorem 22 Suppose the family (π ) is consistent, then the C L mapping Π, such that ( B C )(Π (B) = Π (A)), (15) in which satisfies and A R t such that B = pr 1, (A) is well defined. Furthermore, Π has the following properties: 1. Π preserves finite suprema, and therefore is an isotone mapping from (C, ) to (L, ). 2. If is a nonempty, finite subset of, then the restriction Π C is a (L, )-possibility measure, on ( Xt, C, ). 3. If C is a τ-field on X t, then Π is a (L, )- possibility measure on ( X t, C ). 4. If ( t )((X t, R t ) is a compact topological space ), then Π is P-consistent. 5. Π O is P-consistent. Using theorem 22.5 and theorem 21, we can construct a possibility measure on ( X t, R t ) as follows. heorem 23 Suppose the family (π ) is consistent. hen there exist a complete lattice (L, ), a supremum preserving order-embedding φ from (L, ) to (L, ) and a (L, )-possibility measure Π on ( Xt, R t ), such that Π O = φ (Π O ), in which Π is the mapping constructed in theorem 22. If the mapping Π O satisfies at least one of the sufficient conditions for extendability (E 1 ), (E 2 ), (E 3 ), then one can take (L, ) for (L, ) and 1 L for φ. his theorem can be translated into the following result, which provides the starting point for a measure-theoretic treatment of possibilistic processes. heorem 24 Suppose the family (π ) is consistent. hen there exist a (L, )-possibility space (Ω, R Ω, Π Ω ), where (L, ) is a complete lattice in which (L, ) is embedded using a supremum preserving mapping φ from (L, ) to (L, ), and a family (f t t ) of (L, )-possibilistic variables in the family ((Xt, R t ) t ), with (Ω, R Ω, Π Ω ) as basic space, such that for any x X t, and π f t (x) = (φ π )(x) Π f t ( X t ) = (φ Π )( X t ), for any nonempty, finite subset of. If the mapping Π O, constructed in theorem 22, satisfies at least one of the sufficient conditions for extendability (E 1 ), (E 2 ), (E 3 ), then one can take (L, ) for (L, ) and 1 L for φ. If the field C is a τ-field on X, then there exist a (L, )-possibility space (Ω, R Ω, Π Ω ) and a family (f t t ) of (L, )-possibilistic variables in the family ((X t, R t ) t ) with (Ω, R Ω, Π Ω ) as basic space, such that π is the possibility distribution of f t for any nonempty, finite subset of. From the previous theorem we can immediately derive a possibilistic Daniell-Kolmogorov theorem [Doob, 1967].

Corollary 25 Suppose the family (π ) is consistent. Furthermore, assume that any τ- space (X t, R t ) coincides with a τ-space (X, R) for any t. hen there exist a (L, )-possibility space (Ω, R Ω, Π Ω ), where (L, ) is a complete lattice in which (L, ) is embedded using a supremum preserving mapping φ from (L, ) to (L, ), and a possibilistic process (f t t ) in a *-extension (X, R ) of (X, R), with (Ω, R Ω, Π Ω ) as basic space, such that and π f t X = φ π, Π f t ((X ) ) = (φ Π )(X ), for any nonempty, finite subset of. If the mapping Π O, constructed in theorem 22, satisfies at least one of the sufficient conditions for extendability (E 1 ), (E 2 ), (E 3 ), then one can take (L, ) for (L, ) and 1 L for φ. If the field C of measurable cylinders of (X, R ) is a τ-field on X, then there exist a (L, )-possibilitity space (Ω, R Ω, Π Ω ) and a family (f t t ) of (L, )-possibilistic variables in the τ- space (X, R) with (Ω, R Ω, Π Ω ) as basic space, such that π is the possibility distribution of f t for any nonempty, finite subset of. Note that this is not a perfect analogon of the probabilistic Daniell-Kolmogorov theorem. In case of a noncompact sample space (X, R), we obtain a possibilistic process (f t t ) in a one-point extension (X, R ). Although this sample space (X, R ) is larger than the given τ-space (X, R), we obtain for any nonempty, finite subset of that the restriction π f t X of the joint distribution of the finite subfamily (f t t ) of the possibilistic process (f t t ) to the finite Cartesian power X of the given sample space (X, R) is completely determined by the given distribution π. Furthermore, Π f t ((X ) ) = (φ Π )(X ) = Π f t (X ) for any nonempty, finite subset of. So, by extending the sample space (X, R) to a one-point extension (X, R ), the possibility of a finite Cartesian product of the sample space (X, R ) still equals the possibility of the corresponding Cartesian product of the given space (X, R). In case the given space (X, R) is compact, we have a true analogon of the probabilistic Daniell-Kolmogorov theorem. If the index set is finite, we obtain the following result. Corollary 26 Suppose the family (π ) is consistent. If is finite, then there exist a (L, )-possibility space (Ω, R Ω, Π Ω ) and a family (f t t ) of (L, )-possibilistic variables in the family ((X t, R t ) t ) with (Ω, R Ω, Π Ω ) as basic space, such that π is the possibility distribution of f t for any nonempty subset of. If any τ-space (X t, R t ) coincides with a τ-space (X, R) for any t, this corollary is a special case of the possibilistic Daniell-Kolmogorov theorem, which is a true analogon of its probabilistic counterpart. References [De Cooman and Kerre, 1993] G. de Cooman and E. E. Kerre. Ample Fields. Simon Stevin, 67:235 244, 1993. [Boyen et al., 1995] L. Boyen, G. de Cooman and E.E. Kerre. On the extension of P-consistent mappings. In De Cooman, G. and Ruan, D. and Kerre, E. E., editor, Foundations and Applications of Possibility heory, pages 88 98, Singapore, 1995. World Scientific. [De Cooman, 1993] G. de Cooman. Evaluatieverzamelingen en -afbeeldingen - Een ordetheoretische benadering van vaagheid en onzekerheid [Evaluation Sets and Mappings - An Order-heoretic Approach to Vagueness and Uncertainty]. PhD thesis, Universiteit Gent, Ghent, 1993. [Doob, 1967] J. L. Doob. Stochastic Processes. John Wiley & Sons, New York, 1967. [Kelley, 1959] J. L. Kelley. General opology. D. Van Nostrand, Princeton, New Jersey, 1959. [Wang and Klir, 1992] Z. Wang and G. J. Klir. Fuzzy Measure heory. Plenum Press, New York, 1992. [Wang, 1982] P.-Z. Wang. Fuzzy contactibility and fuzzy variables. Fuzzy Sets and Systems, 8:81 92, 1982. [Wang, 1985] Z. Wang. Extension of possibility measures defined on an arbitrary nonempty class of sets. In Proceedings of the First IFSA Congress, Palma de Mallorca, Spain, 1985. [Willard, 1970] S. Willard. General opology. Addison-Wesley, Reading, Massachusetts, 1970.