Almost-measurability relation induced by lattice-valued partial possibilistic measures

Abstract

Possibilitic measures are set functions which can be taken into consideration as a mathematical tool for uncertainty quantification and processing, alternative to standard probability measures. For a number of practical reasons and restrictions, worth a more detailed investigating are partial possibilistic measures with non-numerical values, e.g. with values in partially ordered sets (POS) in general or in complete lattices in particular. For non-measurable sets, i.e. for the sets outside the domain of the partial possibilistic measure in question, we define their inner and outer measure, applying the basic idea of classical measure theory and approximating these sets, in the best possible way, by their measurable subsets and coverings. Inspired by the notion of symmetric difference, we introduce a lattice-valued metric or distance function and define a set to be almost measurable, if the distance between the value of its inner and outer measure is below a "small" lattice-valued threshold values, so generalizing the idea of measurability in the Lebesgue sense from the standard measure theory. A number of results dealing with the notion of lattice-valued almost-measurability and with the classes of almost measurable sets are stated and proved.