数列和可测函数的几乎收敛性  

The almost convergence of sequence and measurable function

作　　者：杨欣雨 陆艺 石攀岩 周丽珍[1] YANG Xinyu;LU Yi;SHI Panyan;ZHOU Lizhen(School of Mathematical Sciences,Soochow University,Suzhou 213000,Jiangsu Province,China)

机构地区：[1]苏州大学数学科学学院,江苏苏州215031

出　　处：《浙江大学学报（理学版）》2023年第2期131-136,159,共7页Journal of Zhejiang University（Science Edition）

摘　　要：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R^(n)上的Lebesgue测度为基础,建立了R^(n)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R^(n)上几乎处处几乎连续。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 R^(n),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 R^(n).

关 键 词：密度 几乎收敛 几乎连续 

分 类 号：O174.1[理学—数学]