Some properties and convergence theorems of set-valued Choquet integrals

The paper aims at discussing the set-valued Choquet integral, which is the integral of set-valued random variables with respect to capacities. We mainly present representation theorems of the set-valued random variable by using a sequence of Choquet integrable selections and then we investigate some properties of set-valued Choquet integrals, especially subadditive property and inequality of the metric of set-valued Choquet integrals. We also prove Fatouʹs Lemmas, Lebesgue dominated convergence theorem and monotone convergence theorem of set-valued Choquet integrals under the weaker conditions than that in previous works.