关于Lebesgue积分中的极限定理

On the limit theorems in the theory of Lebesgue integral

本文首先给出了控制收敛定理的一个完全独立和更为直接的证明,同时提供了积分与极限可交换次序的一个充要条件,然后利用控制收敛定理来证明Levi渐升列定理,最后还讨论了Levi定理、Fatou引理和Lebesgue控制收敛定理之间的等价关系。

We first give a completely independent and more direct proof for Lebesgue Dom-inated Convergence Theorem and provide a sufficient and necessary condition for commutation of integral and pointwise convergence of a sequence of functions.Then we try to prove Levi Theorem by using Dominated Convergence Theorem.Finally we discuss equivalence relations of Levi Theorem,Fatou Lemma and the Dominated Convergence Theorem.