POSSIBILITY AND NECESSITY INTEGRALS

In this paper, we introduce seminormed and semiconormed fuzzy integral s associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codom ain. In introducing these integrals, the analogy with the classical in troduction of Lebesgue integrals is explored and exploited. It is amon gst other things shown that our integrals are the most general integra ls that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalizati on of Sugeno's fuzzy integrals. A large number of important general pr operties of these integrals is studied. Furthermore, and most importan tly, the combination of seminormed fuzzy integrals and possibility mea sures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied. It is shown that t hese combinations are very natural, and have properties which are anal ogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure- a nd 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 th eory.