The standard techniques of lower and upper approximations, used in
order to define the inner and outer measures given a σ-additive measure, perhaps a probabilistic one, are applied to possibilistic measures. The conditions under which this approach can be reasonable and useful are investigated and the most elernentary properties of the resulting inner and outer possibilistic measures are briefly sketched.
Investigated are possibilistic distributions taking as their values sequences from the infinite Cartesian product of identical copies of a fixed finite subset of the unit interval of real numbers. Uniform and lexicographic partial orderings on the space of these sequences are defined and the related complete lattices introduced. Lattice-valued entropy function is defined in the common pattern for both the orderings, naturally leading to different entropy values for the particular ordering applied in the case under consideration. The mappings on possibilistic distributions with uniform partial ordering under which the corresponding entropy values are conserved as well as approximations of possibilistic distributions with respect to this entropy function are also investigated.
Non-riumerical fuzzy and possibilistic measures taking their values in
partially ordered sets, semilattices or lattices are introduced. Using the operations of supremurn and infimum in these structures, the inner and outer (lower and upper) extensions of the original measures are investigated and defined. The conditions under which the resulting functions -extend conservatively the original ones and possess the properties of fuzzy or possibilistic measures, are explicitly stated and relevant assertions are proved.
Possibilistic measures are usually defined as set functions ascribibg to each subset of the universe of cliscourse a real number from the unit interval and obeying sonie well-kiiown simple conditions. For the number of reasons, as a more realistic version of this model, let us consider partial possibilistic measures defined only for certain subsets and ascribing to them, instead of real numbers, elements from a more general structure. As a rule, a complete lattice will play this role, so let us pick up rather the qualitative and comparative than the quantitative features of particular degrees of possibility. Following the ideas of the standard measure theory, we define the inner and the outer measure induced by the partial latticevalued possibilistic measure in question. A subset of the basic universe is defined as ahnost measurable, if the difference (or rather distance) between the values of the inner and the outer measure ascribed to this set does not exceed, in the sense of the partial ordering relation defined in the used complete lattice, some given threshold value (a “small” fixed element from this lattice). Properties of systems of almost measurable sets are investigated in greater detail and some assertions related to them are introduced.
Elementary random events possibly favorable to a random event are defined as those elementary random events for which we are not able to prove or deduce, within the limited framework of a decision procedure being at our disposal, that they are not favorable to the random event in question. Under some conditions, probabilities of the sets of possibly favorable elementary random events induce uniquely a possibilistic measure on the system of all subsets of the universe of elementary random events under consideration. Moreover, each possibilistic measure defined on this power-set can be obtained, or at least approximated to the degree of precision a priori given, in this way. Some results concerning the combinations of decision systems and decision systems induced by random variables are introduced and proved.