1、riemann積分可積性理論探討riemann積分可積性理論探討摘要本文較為系統(tǒng)地討論了積分可積性理論：通過分析諸多積分概念的共性，抽象定義了積分，詳細討論了其可積性理論，得出了可積函數(shù)類 . 從極限理論出發(fā)定義了正規(guī)函數(shù)，其可積性理論統(tǒng)1 了積分的 3 個常用的充分條件，并用理論和有限覆蓋定理予以證明 . 通過定義 0 測度集給出了 可積函數(shù)的 特征，討論了其幾乎處處連續(xù)與極限存在的關(guān)系，從而得到了從函數(shù)可積性到連續(xù)性，從連續(xù)性到極限存在性的函數(shù)特性理論，即可積函數(shù)中極限的幾乎處處存在與幾乎處處連續(xù)是等價的，得出比正規(guī)函數(shù)更加寬泛的統(tǒng) 1條件，得出了有界變差函數(shù)是可積函數(shù)的結(jié)論 . 通過定義

2、多維 0測度集將 可積函數(shù)的 特征擴展到多維情形，同樣統(tǒng)1 了多維情形的充分條件，建立了多維情形的可積性理論. 關(guān)鍵詞積分；可積條件；正規(guī)函數(shù)；幾乎處處連續(xù)；0 測度集；極限the study of the the integrability ofriemannsintegral theoryabstractthis paper discusses the integ rability of riemann s integral theory systematically: by analyzing the common characters of a lot of integral calc

3、ulus, it abstractsthe concept of riemann integral and discusses itsintegrability of riemann s integral theory and then gets integrable functions. it defines the regulated functionfrom the theory of extreme limit,the integrability theoryof the regulated function unifies the three commonsufficient con

4、ditions of the integral, then the paperproves that with the darboux theory and heine-boreltheory. by getting lebesgue characteristic of integrablefunction of riemann from the definition of gather zeromeasure, discussing the relation between almostcontinuous everywhere and existent of limit, it gets

5、thetheory which is from the function integrability to theconsecution and from consecution to the limitexistence .i.e. the almost limit existence is equal tothe almost continuous everywhere in the integrablefunction of riemann. it also gets a unified conditionwhich has a wider range than regulated fu

6、nction and comesto the conclusion that the function of bounded variationis 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 integ

7、ral; integrable condition; regulated function;almost continuous everywhere; gather zero measure; extreme limitriemann積分可積性理論探討摘要本文較為系統(tǒng)地討論了 積分可積性理論：通過分析諸多積分概念的共性，抽象定義了 積分，詳細討論了其可積性理論，得出了可積函數(shù)類 . 從極限理論出發(fā)定義了正規(guī)函數(shù)，其可積性理論統(tǒng)1 了 積分的 3 個常用的充分條件，并用理論和有限覆蓋定理予以證明 . 通過定義 0 測度集給出了可積函數(shù)的特征，討論了其幾乎處處連續(xù)與極限存在的關(guān)系，從而得到了從函數(shù)

8、可積性到連續(xù)性，從連續(xù)性到極限存在性的函數(shù)特性理論，即可積函數(shù)中極限的幾乎處處存在與幾乎處處連續(xù)是等價的，得出比正規(guī)函數(shù)更加寬泛的統(tǒng) 1 條件，得出了有界變差函數(shù)是可積函數(shù)的結(jié)論 . 通過定義多維 0 測度集將可積函數(shù)的特征擴展到多維情形，同樣統(tǒng)1了多維情形的充分條件，建立了多維情形的可積性理論. 關(guān)鍵詞積分；可積條件；正規(guī)函數(shù)；幾乎處處連續(xù)；0 測度集；極限the study of the the integrability ofriemannsintegral theoryabstractthis paper discusses the integrability of riemann i

9、ntegral 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 getssintegrable functions. it defines the regulated function from the theory of extreme

10、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

11、measure, discussing the relation between almost continuous everywhere and existent of limit, it gets thetheory which is from the function integrability to theconsecution and from consecution to the limitexistence .i.e. the almost limit existence is equal tothe almost continuous everywhere in the int

12、egrablefunction of riemann. it also gets a unified conditionwhich has a wider range than regulated function and comesto the conclusion that the function of bounded variationis the integrable function of riemann. it expands lebesgue characteristic of integrable function of riemann through the definition of gather zero me