Lebesgue积分的产生和意义

论述Lebesgue积分产生的背景和思想形成过程,重点介绍直观、想象、类比、猜测等思维形式在Lebesgue积分产生过程中所发挥的作用．     

瑕积分的收敛性与Lebesgue可积性

瑕积分的收敛性与Lebesgue可积性之间具有密不可分的关系.首先详细证明了瑕积分收敛定理,然后说明无穷积分的收敛定理与瑕积分的收敛定理是等价的,最后利用两条定理的等价性给出一个应用.     

从泛函延拓的视角看Lebesgue积分和Riemann积分

约定1〈p〈∞,定义空间Cp[a,b],证明Cp[a,b]是Lp[a,b]的子空间.利用Lebesgue积分和Riemann积分在Lp[a,b]和Cp[a,b]上分别定义线性泛函L和R,证明二者有界且有相等范数.利用Taylor定理和已得结论证明L是R从Cp[a,b]到Lp[a,b]的唯一保范延拓.     

一道Lebesgue积分题目的注记

设E为n维欧氏空间Rn的可测子集,m E+∞,f(x)为E上的非负可测函数,并记Ek=E{x|k≤f(x)k+1}(k=0,1,2,…),利用Lebesgue积分的性质,通过构造反例,指出"级数∑k km Ek收敛"并不是f(x)在E上Lebesgue可积的充分条件.进一步,通过增加条件"f在E上a.e.有限",得到相应的充分必要条件.     

Riemann—Lebesgue定理及其应用

本文利用实变函数方法详细证明了Riemann-Lebesgue定理在开集上成立,并且给出了其应用.     

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.     

Riemann积分的历史与发展

总结Riemann积分的发展历史、基本思想及其各种推广,介绍Riemann积分的局限性和缺陷,以及由此推动产生Lebesgue积分的过程.     

Lebesgue积分列的等度绝对连续性

给出了Lebesgue积分列的等度绝对连续的一个充分必要条件.     

关于瑕积分与Lebesgue积分之关系的一种证法

本给出了函数f(x)的瑕积分绝对收敛时．必定Lebesgue可积的一种证明方法．     

Lebesgue积分的另一种定义

用可数分法代替有限分法,可去掉函数有界和定义域测度有限的限制,一次性地作出函数Lebesgue可积的定义并得出可积的充要条件.     

A random approach to the Lebesgue integral

We construct an integral of a measurable real function using randomly chosen Riemann sums and show that it converges in probability to the Lebesgue integral where this exists. We then prove some conditions for the almost sure convergence of this integral.     

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.     

Riemann integrability and Lebesgue measurability of the composite function

If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c²-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}.     

关于勒贝格积分的一个注记

本文给出了函数 f(x)的瑕积分绝对收敛时必定 Lebesgue可积的一种证明方法     

G可积函数的Lebesgue可测性

Botsko在连续和可导的知识基础上推广了Riemann积分,得到了一种新的积分,称为G积分．G积分既不同于Riemann积分也不同于Lebesgue积分．本文通过对G积分的研究,得到了G可积函数一定Lebesgue可测,从而有界G可积函数一定Lebesgue可积;同时我们还证明了这两个积分值相等．     

Lebesgue积分的产生及其影响(迎接ICM2002特约文章)

这是一篇由法国科学院院士 Ｋａｈａｎｅ为纪念Ｌｅｂｅｓｇｕｅ积分 １００周年而写的文章,原文发表在《巴黎科学院通报》上,文章讲述了Ｌｅｂｅｓｇｕｅ积分创立的历史背景及其影响,经作者同意,由武汉大学范爱华教授翻译成中文在本刊发表．     

The Lebesgue differentiation theorem revisited

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on ${\mathbb{R}}^n$ in terms of approximate continuity associated to an expansive linear map.     

On Lebesgue Functions of Uniformly Bounded Orthonormal Systems

A complement to A. M. Olevskii's fundamental inequality on the logarithmic growth of Lebesgue functions of an arbitrary uniformly bounded orthonormal system on a set of positive measure is made. Namely, the index where the Lebesgue functions have growth slightly weaker than logarithmic can be chosen independently of the variable. The theorem proved in this paper improves a result established earlier by the author.