Probability capacities in \(\mathbb{R}^d\) and the Choquet integral for capacities (Q1884468)

Choquet capacity is a generalization of Borel measures on \(\mathbb{R}^d\). The authors try to extend some notions related to Borel measures to Choquet capacity. For a Choquet capacity \(T\) on \(\mathbb{R}^d\), the support of \(T\) is defined naturally and \(T\) is called probability capacity if \(T(\text{supp}\,T)=T(\mathbb{R}^d)=1\). The Choquet integral for any measurable function \(f\) is defined by \[ \int_A f \,dT=\int_0^\infty T(\{x \in A:f(x)\geq t\}) \,dt. \] As an example, for a finite set \(A=\{(x_1,t_1),\dots,(x_k,t_k)\} \subset \mathbb{R}^r\times \mathbb{R}^+\), if \(T_A\) is the capacity defined by \(T_A(B)=\max\{t_i: x_i \in B\}\), then \[ \int f \,dT_A= \sum_{i=0}^{k-1} (\alpha_{i+1}-\alpha_i) \max\{t_{n_j}:j=i+1,\dots,k\}, \] where \(\alpha_i=f(x_{n_i})\) is arranged as an increasing order. Some other examples of Choquet integrals are also presented. Furthermore, capacity in the sense of Graf is introduced and its difference from the Choquet capacity is explained by an example.