非负可测函数L积分的定义及其等价性  

The definitions of Lebesgue integral of non-negative measurable functions and their equivalence

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作  者:张永锋[1] 

机构地区:[1]咸阳师范学院数学系,陕西咸阳712000

出  处:《纺织高校基础科学学报》2007年第3期241-245,共5页Basic Sciences Journal of Textile Universities

摘  要:勒贝格积分理论是实变函数的核心.关于勒贝格积分有不止一种的定义方法,研究勒贝格积分的定义方法,证明勒贝格积分不同定义的等价性,对于简化和统一勒贝格积分定义,从不同角度理解和掌握勒贝格积分概念以及灵活运用勒贝格积分具有重要意义.采用分割函数定义域和分割函数值域以及用简单函数列逼近等方法,研究了非负可测函数勒贝格积分的定义,给出了非负可测函数勒贝格积分的4种定义,并且仅从所给定义出发,比较初等地证明了它们的等价性.The theorem of Lebesgue integral is one of the most important parts in real variable function. About Lebesgue integral, there are different definition modes. For simplifying and unifying the definitions,and for understanding and mastering the concepts of Lebesgue integral from different angle,it is very significant to study how to define Lebesgue integral,and to prove the equivalence of different defintions. In this paper, the definition modes of Lebesgue integral of non-negative measurable functions are studied by the way of cutting defining and valued fids,approaching with simple function series. The four definitions of Lebesgue integral of non-negative measurable functions are given. Furthermore, their equivalence properties are proved by elementary knowledge.

关 键 词:非负可测函数 勒贝格积分 定义 等价性 

分 类 号:O174.1[理学—数学]

 

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