Possibility and necessity integrals

Abstract

In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Lebesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno's fuzzy integrals. A large number of important general properties of these integrals is studied. Furthermore, and most importantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied. It is shown that these combinations are very natural, and have properties which are analogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure- and integral-theoretic account of possibility and necessity theory, in very much the same way as the theory of Lebesgue integration provides a proper framework for a unifying and formal account of probability theory.