Real-valued Choquet integrals for set-valued mappings

• A new kind of Choquet integrals for set-valued mappings is introduced.
• Elementary properties and convergence theorems are shown.
• In the case of the monotone convergence theorem of the nonincreasing sequence, the integrands must be closed.
• This kind of Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.

In this paper a new kind of real-valued Choquet integrals for set-valued mappings is introduced, and some elementary properties of this kind of Choquet integrals are studied. Convergence theorems of a sequence of Choquet integrals for set-valued mappings are shown. However, in the case of the monotone convergence theorem of the nonincreasing sequence of Choquet integrals for set-valued mappings, we point out that the integrands must be closed. Specially, this kind of real-valued Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.