Monotone set-valued measures: Choquet integral, $f$-divergence and Radon-Nikodym derivatives

Divergence as a degree of the difference between two data is widely used in the classification problems. In this paper, $f$-divergence, Hellinger divergence and variation divergence of the monotone set-valued measures are defined and discussed. It proves that Hellinger divergence and variation divergence satisfy the triangle inequality and symmetry by means of the set operations and partial ordering relations. Meanwhile, the necessary and sufficient conditions of Radon-Nikodym derivatives of the monotone set-valued measures are investigated. Next, we define the conjugate measure of the monotone set-valued measure and use it to define and discuss a new version $f$-divergence, and we prove that the new version $f$-divergence is nonnegative. In addition, we define the generalized $f$-divergence by using the generalized Radon-Nikodym derivatives of two monotone set-valued measures and examples are given. Finally, some examples are given to illustrate the rationality of the definitions and the operability of the applications of the results.