一致收敛的几种拓广方式及其相互联系

On Extension Methods of Uniform Convergence and Their Interconnections

给出了近一致收敛,几乎处处收敛,依测度收敛的简单、直观而严格的集合序列描述,应用集合序列描述大幅度简化了著名的Eropo(?)、Lebesgue等系列定理的证明过程.将Riesz定理从研究依测度收敛与几乎处处收敛的关系的研究方法移植到研究依测度收敛与近一致收敛的关系,并突破mE<+∞的条件限制,证明了移植后的Riesz定理之逆定理.

This paper gives a description of the set sequences,which satisfies nearly uniform convergence, almost everywhere convergence, or convergence in measure. The application of a collection of sequence greatly simplifies the proof of series theorems such as known EropoB and Lebesgue theorem. It expands the Riesz Theorem from ＂studying the relationship of convergence in measure and almost everywhere convergence＂ to ＂studying the relationship between convergence in measure and near uniform convergence ＂ ,and breaks through the condition of mE 〈 ＋ ∞ , proves inverse theorem of Riesz theorem after the Extension .