Approximations of lattice-valued possibilistic measures

Lattice-valued possibilistic measures, conceived and developed in more detail by G.*De Cooman in 1997 [3], enabled to apply the main ideas on which the real-valued possibilistic measures are founded also to the situations often occurring in the real world around, when the degrees of possibility, ascribed to various events charged by uncertainty, are comparable only quantitatively by the relations like "greater than" or "not smaller than", including the particular cases when such degrees are not comparable at all. The aim of this work is to weaken the demands imposed on possibilistic measures in other direction: the condition that the value ascribed to the union of two or more events (taken as subsets of a universe of discourse) is identical with the supremum of the values ascribed to particular events is weakened in the sense that these two values should not differ "too much" from each other, in other words, that their (appropriately defined) difference should be below a given "small" threshold value. This idea is developed, in more detail, for the lattice-valued possibility degrees, resulting in the notion of lattice-valued quasi-possibilistic measures. Some properties of these measures are investigated and relevant mathematically formalized assertions are stated and proved.