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．     

On the limits of generalization of the Kolmogorov integral

The generalization of the Kolmogorov integral to functions with values in a Banach space is considered. It is proved that the resulting integral turns out to be essentially more general than the Bochner integral and is exactly equivalent to an integral of McShane type, whose definition requires that the scaling function be measurable.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 258–272.Original Russian Text Copyright © 2005 by A. P. Solodov.This revised version was published online in April 2005 with a corrected issue number.     

Almost every 2-SAT function is unate

Bollobás, Brightwell and Leader [2] showed that there are at most 2-SAT functions on n variables, and conjectured that in fact the number of 2-SAT functions on n variables is . We prove their conjecture. As a corollary of this, we also find the expected number of satisfying assignments of a random 2-SAT function on n variables. We also find the next largest class of 2-SAT functions and show that if k = k(n) is any function with k(n) < n ^1/4 for all sufficiently large n, then the class of 2-SAT functions on n variables which cannot be made unate by removing 25k variables is smaller than for all sufficiently large n.     

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.     

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.     

Aczel积分形的不等式推广

在文献［１］推广了有限和式的Ａｃｚｅｌ不等式，本文继续给出积分形式的Ａｃｚｅｌ不等式。     

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.     

关于江泽坚等著《实变函数论》的一个小瑕疵

指出江泽坚等合编教材《实变函数论》[2-3]中的一处瑕疵.这个瑕疵起始于第二版,在第三版时又没更正,时间已有22年(1994-2016).鉴于此教材影响大,使用面广,我们特写此注记将此瑕疵指出,与作者和读者商榷.     

Weighted Estimates for the Integral Operators with Monotone Kernel on a Half-Axis

We give criteria for the boundedness of integral operators with nonnegative monotone kernel in the weighted Lebesgue spaces on a half-axis.     

判别函数可积性的一个新的重要命题

在抽象测度空间中,用可测集EK去逼近集E的办法,从函数f在E上的可测性去推f在E上的可积性,是判别函数可积性的一个新的重要命题,但[2]在证明这一命题时有误.本文作了更正,并从距离空间中的积分推广到抽象测度空间中的积分.     

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．     

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.     

多重积分的积分中值定理

利用开区域的道路连通性和一元连续函数的介值定理,在L ebesgue积分意义下证明了多重积分的积分中值定理.     

计算柱面上对面积的曲面积分的一种新方法

给出了利用对弧长的曲线积分计算柱面上对面积的曲面积分的一种新方法,其计算公式为∫∫_Σf(x,y,z)dS=∫_(L*)ds∫z_1(x,y) z_2(x,y)f(x,y,z)dz,其中积分曲面Σ为垂直于xoy坐标面的柱面片,L*为Σ在xoy坐标面上的投影曲线(平面曲线),z=z1(x,y),z=z2(x,y)分别为过Σ的下边界曲线和上边界曲线的任一不同于Σ的曲面的方程.     

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.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Not every -set is perfectly meager in the transitive sense

We prove the following theorems:

1.

It is consistent with ZFC that there exists a - set which is not perfectly meager in the transitive sense.

2.

Every set which is perfectly meager in the transitive sense has the property.

3.

The product of two sets perfectly meager in the transitive sense has also that property.