First results for a mathematical theory of possibilistic Markov processes
Hj Janssen, Gert de Cooman, Etienne E. Kerre · 1996
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 ′ ⋐ 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 ′ ⋐ 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 Daniell-Kolmogorov [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.