Generalizations and Extensions of Lattice-Valued Possibilistic Measures, Part II

In this technical report, the systematic investigation of lattice-valued possibilistic measures, opened in the first part of this report (cf. Technical Report No. 952, ICS AS CR, December 2005) is pursued and focused towards possibilistic variants of the notions of outer (upper) and lower (inner) possibilistic measures induced by a given partial possibilistic measure. The notion of Lebesgue measurability for possibilistic measures is defined and it is generalized to the notion of almost-measurability in the sense that the values of inner and outer measure ascribed to a set are not identical, as demanded in the case of Lebesgue measurability, but these two values do not differ ”too much” from each other in the sense definable within the lattice structure. In the rest of this report, the probabilistic model of decision making under uncertainty is modified to the case of lattice-valued possibilistic measures, so arriving at the notion of possibilistic decision function. The Bayesian and the minimax principles when quantifying the qualities of particular possibilistic decision functions are also analyzed and applied to some cases, e.g., to the case of possibilistic decision functions for state identification.