Lebesgue theorems in non-additive measure theory

In this paper, the sufficient and necessary conditions for different kinds of Lebesgue theorem in non-additive measure theory are presented, respectively. The equivalence among the Lebesgue theorem, the Monotone convergence theorems of fuzzy and of Choquet integral are shown. As direct results of the Lebesgue theorems, the necessary conditions for Egoroff's theorem are given.