Riemann积分可积性理论探讨

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翻新时间：2023-03-11

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Riemann积分可积性理论探讨

摘要

本文较为系统地讨论了积分可积性理论：通过分析诸多积分概念的共性，抽象定义了积分，详细讨论了其可积性理论，得出了可积函数类.从极限理论出发定义了正规函数，其可积性理论统1了积分的3个常用的充分条件，并用理论和有限覆盖定理予以证明.通过定义0测度集给出了可积函数的特征，讨论了其几乎处处连续与极限存在的关系，从而得到了从函数可积性到连续性，从连续性到极限存在性的函数特性理论，即可积函数中极限的几乎处处存在与几乎处处连续是等价的，得出比正规函数更加宽泛的统1条件，得出了有界变差函数是可积函数的结论.通过定义多维0测度集将可积函数的特征扩展到多维情形，同样统1了多维情形的充分条件，建立了多维情形的可积性理论.

关键词积分；可积条件；正规函数；几乎处处连续；0测度集；极限

The study of the the integrability of Riemann’s

integral Theory

ABSTRACT

This paper discusses the integrability of Riemann’s integral theory systematically: By analyzing the common characters of a lot of integral calculus, it abstracts the concept of Riemann integral and discusses its integrability of Riemann’s integral theory and then gets integrable functions. It defines the regulated function from the theory of extreme limit,The integrability theory of the regulated function unifies the three common sufficient conditions of the integral, then the paper proves that with the Darboux theory and Heine-Borel theory. By getting Lebesgue characteristic of integrable function of Riemann from the definition of Gather zero measure, discussing the relation between almost continuous everywhere and existent of limit, it gets the theory which is from the function integrability to the consecution and from consecution to the limit existence .i.e. the almost limit existence is equal to the almost continuous everywhere in the integrable function of Riemann. It also gets a unified condition which has a wider range than regulated function and comes to the conclusion that the function of bounded variation is the integrable function of Riemann. It expands Lebesgue characteristic of integrable function of Riemann through the definition of gather zero measure and builds up the theory of many integral calculus.

Keywords: Riemann integral; Integrable condition; Regulated function; Almost continuous everywhere; Gather zero measure; Extreme limit