关于叶果洛夫定理和Lebesgue定理的扩展

Expansion about EropoB and Lebesgue

叶果洛夫定理和Lebesgue定理中共有的条件"fm(m=1,2,…)是E上几乎处处有限的可测函数"可以减弱为"fm(m=1,2,…)是E上的可测函数";"f有限a.e于E"可减弱为"f有限a.e于E或f无限a.e于E"。给出在这种条件减弱的情况下三种收敛的关系。

The strength of hypothesis in both EropoB Theorem and Lebesgue Theorem-＂ The function fro （m = 1, 2… ） which is finite almost everywhere on E（ abbreviated to fm is finite a. e on E）is measurable. ＂-can be weakened as ＂fm （ m = 1,2…） is a measurable function on E＂ ; The strength of the hypothesis＂f is finite a. e on E＂ can be weakened as＂ f is finite a. e on E orf is infinite a. e on E＂. Under this condition that the strength of the hypothesis is weakened, the relationships are analyzed among these three kinds of convergence.