Conditional Choquet Expectation

Capacity plays an important role in many areas. A capacity is usually studied under the assumption that it is submodular. In this article, we first present several inequalities for the submodular capacity and state some special submodular capacities. Second, we introduce the concept of conditional Choquet expectation which is the conditional expectation with respect to submodular capacity, and then we prove some properties of this concept, which include positive homogeneity, translation invariance, asymmetry, subadditivity, and monotonicity. Finally, we state that Cr inequality, Schwarz inequality, Ho¨$\ddot{o}$lder’s inequality, Minkowski inequality, and Jensen’s inequality still hold for this kind of conditional Choquet expectations.