Non-discrete k-order additivity of a set function and distorted measure

Abstract In this study, in a formulaic manner, we generalize the concept of the k-order additivity of a set function. First, we discuss the Mobius transform for a non-discrete set function. Next, we generalize the definition of the k-order additivity of a set function using the Mobius transform and provide the equivalent conditions for the k-order additivity. Furthermore, we consider the k-order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is k-order additive if and only if the distortion function is a polynomial of k-th order.