PAN-INTEGRALS WITH RESPECT TO IMPRECISE PROBABILITIES

As a unified generalization of both the Lebesgue integral and the Suge no integral, pan-integrals were introduced for nonnegative measurable functions with respect to fuzzy measures. In this paper, the concept o f a pan-integral is generalized for nonnegative measurable functions w ith respect to monotone set functions, which include coherent imprecis e probabilities. An inequality is established between two functionals on the class of all nonnegative bounded measurable functions, the natu ral extension of a monotone lower probability and the corresponding pa n-integral whose operators are the common addition and common multipli cation. We also discuss the necessary and sufficient conditions for th e equality between these two functionals. By using a new concept of pa ra-additivity, we derive a necessary and sufficient condition for the equality when the lower probability is restricted to be a belief measu re.