First results for a mathematical theory of possibilistic 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 ø - field to an indexed family of ø -fields, without imposing an ordering on the index set. We also introduce the notion `measurable cylinder ' and show that any product ø-field can be generated by its associated field of measurable cylinders. Furthermore, we introduce and study the notions `ø -subspace', `extension of a ø-space' and `one-point extension of a ø-space'. Using these notions, we prove that for any family of possibility distributions (ß T 0 j ; ae T 0 b T ), satisfying a natural consistency condition, a family (f t j t 2 T ) of possibilistic variables can be constructed such that the possibilistic variable \\Theta t2T 0 f t (with ; ae T 0 b T ) has ß T 0 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...