(L)积分取极限的一个充要条件

A Few Sufficient and Necessary Conditions for the Limits of an(L)Integral

可测函数列 fn(x)的 (L)积分取极限 (即limn→∞ ∫Efn(x)dx) ,是研究可测函数列积分的一种重要方法。对文献 [1]给出的积分号与极限号可交换的一个定理 ,改变了原定理的一个条件 ,作出了简化的证明 ,并得到了积分号下取极限以及函数列具有等度的绝对连续积分的两个充要条件。

The method of limiting integral of measurable function's sequence is a n important key to investigating the integral of measurable function's sequenc e. In general, the integral sign and the limit sign cannot be exchanged. On the cont rary, there are several theorems in [1], which show that they can under some c on ditions. In this paper, the author improves one of the previous theorems' conditions and presents a simplified proof of the theorem.Several sufficient an d necessary conditions are obtained for gaining limit in the integral sign,and equal and absolutely continuous integral of function sequence.