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

In this study, we generalize the concept of the k -order additivity of a set function. First, we discuss the Möbius transform for a non-discrete set function. Next, we generalize the definition of the k -order additivity of a set function using the Möbius 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. [ABSTRACT FROM AUTHOR]