黎曼可积函数列的极限理论及应用

对黎曼可积函数列的极限函数的可积性进行讨论．运用黎曼积分自身的理论依次证明了：一致收敛函数列的极限函数的黎曼可积性,黎曼积分下的控制收敛定理和广义积分下的控制收敛定理。并给出了一些应用例子．     

黎曼积分的完备化

综述了黎曼可积函数的基本特征,并指出黎曼可积函数列的极限运算在积分意义下是不封闭的.在构造了完备化空间之后,证明了该空间就是勒贝格可积函数空间,从而说明了黎曼积分的完备化形式是勒贝格积分.     

从积分概念的发展看数学分析基本概念的发展(续2)

全文分四部分概述了积分概念的发展历史,此为第三部分,主要介绍黎曼积分.从傅里叶的影响开始,详细讨论关于函数概念大争论的情况,特别是狄利克雷对于傅里叶定理的证明,及其对于黎曼积分的影响.介绍黎曼是怎样区别于柯西的.也详细介绍魏尔斯特拉斯所领导的分析的算术化对于整个数学的发展的影响.     

复合函数黎曼可积性的讨论

本文利用有界函数黎曼可积的充要条件讨论了某些复合函数的黎曼可积性,给出了外函数黎曼可积,内函数连续,复合函数不一定黎曼可积的例子．     

关于Banach-值函数的Henstock积分

讨论了Banach-值函数的H enstock积分与P ettis积分的关系,证明了若f是几乎可分值且弱有界的,则f是H enstock可积的当且仅当f是P ettis可积的.     

关于反常积分的一个注记

本文讨论无穷限反常积分∫_a~(+∞)f(x)dx收敛的必要条件.首先考虑黎曼积分意义下该反常积分收敛的必要条件,其结果包含了刘妮、刘卫江的工作(见《高等数学研究》,17卷,6期,6-9页,2014年11月).然后研究勒贝格积分意义下∫_a~(+∞)f(x)dx存在的必要条件.     

时标上区间值函数的若干积分不等式

引入了时标上区间值函数黎曼◇_α-积分的概念,讨论了该积分的基本性质.利用区间分析及时标积分理论,得到了区间值函数黎曼◇_α-积分形式的Jensen不等式、H?lder不等式和Minkowski不等式,推广了现有文献的相关结果.     

函数的原函数存在与黎曼可积的关系

本文讨论原函数存在与黎曼可积之间的联系与区别,通过列举具体的函数来说明函数的原函数存在与黎曼可积是相互独立的概念,两者之间是互不蕴含的关系.     

函数列的黎曼积分极限定理的应用

利用函数列的极限理论方法,研究函数列积分极限中积分和极限可交换次序的问题.对一致收敛的可积函数列给出积分的极限定理,对一致有界局部一致收敛函数列给出积分控制收敛定理,通过大量实例表明该理论的意义所在.     

关于牛顿—莱布尼兹公式

微积分学中一个重要的命题指出:设函数f在闭区间[a,b]上黎曼可积,F在[a,b]上连续且除有限多个点外F′(x)=f(x),则牛顿—莱布尼兹公式成立.文献[1]提出如下问题:若F′(x)=f(x)不成立的点是无限集E,上述结论如何?本文证明当E的聚点集有限时,牛顿—莱布尼兹公式成立;当E的聚点集无限时,反例说明结果是否定的.     

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C^∞ on (0,1] yet fails to be first-return integrable.     

B-值函数Riemann-Stieltjes积分

引入数值函数关于B-值函数的R-S积分,研究了此类积分的性质及向量值R-S积分存在的几个充分条件,并给出了积分的收敛定理.     

向量值函数的Riemann-Stieltjes积分

引入向量值函数关于实值函数的Riemann-Stieltjes积分,给出了向量值Riemann-Stieltjes可积的充要条件,并讨论了积分的收敛定理.     

Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter

In this paper, we obtain a new formula for the representation of the Riemann-Stieltjes integral of a continuous function in terms of the passage to the limit with respect to the parameter in a Riemann integral depending on this parameter. The derivation of this formula is based on the study of the functional properties of the solution of the auxiliary difference equation of first order representing the weighted first difference of a given function in the form of a simple first difference of an unknown function. The result obtained can be used for the analytic and approximate calculation of Stieltjes integrals.     

Consistent first-return Riemann sums for Lebesgue integrals

U. B. Darji and M. J. Evans [1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function on the interval [0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a first-return algorithm based on a sequence {x _n} which is dense in [0,1]. Here we show that if the same is true for every rearrangement of {x _n}, then the function must be equal almost everywhere to a Riemann integrable function. This revised version was published online in June 2006 with corrections to the Cover Date.     

多元Riemann可积函数的特征与完备化

给出了多元Riemann可积函数的基本特征,证明了多元Riemann可积函数空间的完备化是Lebesgue积分空间.     

紧Hausdorff测度空间上的Riemann积分理论

在紧Hausdorff测度空间上建立了Riemann型的积分理论,证明了函数可积的充要条件是该函数几乎处处连续.提出了Riemann积分的可计算性概念,证明了Riemann积分是可计算的当且仅当积分域可以度量化.     

实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分

本文首先建立了实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分,并讨论了集值Riemann-Stieltjes积分的性质,给出了集值Riemann-Stieltjes可积的充要条件,最后引入了集值Riemann-Stieltjes随机积分．     

On the Henstock-Kurzweil Integral for Riesz-Space-Valued Functions Defined on Unbounded Intervals

In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.