Mathematics and Computer Science |
|
|
|
|
The almost convergence of sequence and measurable function |
Xinyu YANG(),Yi LU,Panyan SHI,Lizhen ZHOU() |
School of Mathematical Sciences,Soochow University,Suzhou 213000,Jiangsu Province,China |
|
|
Abstract To generalize the concept of classic limit of sequence, this paper introduces the definition of almost convergent sequence and proves several important properties together with a necessary and sufficient condition of almost convergence, hence building an equivalent relation between almost and strictly convergence. Moreover, based on the Lebesgue measure and by introducing the conception of density of subsets on ,we also provide the definition of almost convergence of measurable functions, including some properties and a basic theorem of almost convergence similar to sequence. Then we introduce the definition of almost continuous function. At last, based on the Lebesgue differential theorem, it is proved that any measurable function is almost continuous, almost everywhere on .
|
Received: 07 April 2022
Published: 21 March 2023
|
|
Corresponding Authors:
Lizhen ZHOU
E-mail: meiyang010420@163.com;zhoulizhen@suda.edu.cn
|
数列和可测函数的几乎收敛性
对经典的数列极限进行了推广,通过引入几乎收敛的定义,证明了几个重要性质以及数列几乎收敛的充分必要条件,建立了几乎收敛与严格收敛之间的等价关系。以上的Lebesgue测度为基础,建立了子集的密度概念,引入了可测函数几乎收敛的定义,证明了与数列几乎收敛平行的若干性质,以及函数几乎收敛基本定理。给出了函数几乎连续的定义,利用Lebesgue微分定理,证明了任意可测函数在上几乎处处几乎连续。
关键词:
密度,
几乎收敛,
几乎连续
|
|
[1] |
史恩慧. 数列几乎收敛及其教学[J]. 高等数学研究, 2012, 15(5): 48-49. DOI:10.3969/j.issn.1008-1399. 2012.05.027 SHI E H. A generalization of sequence limit[J]. Studies in College Mathematics, 2012, 15(5): 48-49. DOI:10.3969/j.issn.1008-1399.2012.05.027
doi: 10.3969/j.issn.1008-1399.2012.05.027
|
|
|
[2] |
潘承洞, 潘承彪. 素数定理的初等证明[M]. 2版. 哈尔滨: 哈尔滨工业大学出版社, 1988. PAN C D, PAN C B. The Elementary Proofs of the Prime Number Theorem[M]. 2nd ed. Harbin: Harbin Institute of Technology Presse, 1998.
|
|
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
|
Shared |
|
|
|
|
|
Discussed |
|
|
|
|