Convergence in, or with respect to, s-additive measure, in particular, convergence in probability, can be taken as an important notion of the standard measure and probability theory, and as a powerful tool when analyzing and processing sequences of subsets of the universe of discourse and, more generally, sequences of real-valued measurable functions defined on this universe. Our aim is to propose an alternative of this notion of convergence supposing that the measure under consideration is a (complete) non-numerical and, in particular, lattice-valued possibilistic measure, i.e., a set function obeying the demand of (complete) maxitivity instead of that of s-additivity. Focusing our attention to sequences of sets converging in a lattice-valued possibilistic measure, some more or less elementary properties of such sequences are stated and proved.
Given a possibilistic distribution on a nonempty space Ω with possibility degrees in a chained complete lattice, the lattice-valued entropy function for such distribution is defined as the expected value (in the sense of Sugeno possibilistic integral) of the lattice-valued function ascribing to each ωeΩ the possibilistic measure of its complement Ω-{ω}.
However, such an entropy function seems to be little sensitive or flexible in the sense that it ascribes the same and supremum value to a rather wide class of different lattice-valued possibilistic distributions so that the choice of the most adequate, in a sense, distribution (for the purposes of decision making under uncertainty) is rather limited. In this paper, we propose and analyze a refined version of this entropy, which splits the wide class of possibilistic distributions mentioned above into a rich spectre of narrower classes of distributions to which different but mutually comparable values of the refined entropy function are ascribed.