毕业论文勒贝格积分

本科毕业论文勒贝格积分中三大极限定理的等价关系研究及应用I摘要勒贝格积分中的三大极限定理是勒贝格积分极限理论体系的中心内容,包括勒贝格控制收敛定理、列维定理与Fatou引理.这三大极限定理在分析学中占有很重要的地位.本文主要针对勒贝格积分三大极限定理的等价关系及应用两方面进行阐述.先证明Fatou引理,然后用Fatou引理证明列维定理,再用列维定理证明列贝格控制收敛定理,最后用勒贝格控制收敛定理证明Fatou引理,通过这一循环过程,即可得到三大极限定理是相互等价的结论；然后对三大极限定理在积分与极限交换运算中的应用和非正函数中的应用等内容进行了探讨.关键词:极限定理；等价关系；应用AbstractThethreelimittheoremsinLebesgueintegralisthecentercontentoflimittheorysystem,whichincludeLebesguecontro1convergencetheorem,LeviandFatoulemma.Thethreelimittheoremsplayaveryimportantroleinanalysis.ThethreelimittheoremofLebesgueintegralaremainlydiscussedfromtheequivalencerelationandapplicationsaspects.Firstly,theFatoulemmaisproved.Secondly,theFatoulemmaisusedtoproveLevitheorem.Thirdly,theLevitheoremisusedtoproveLebesguecontro1convergencetheorem.Finally,theLebesguecontro1convergencetheoremisusedtoproveFatoulemma.Sotheconclusionisthethreelimittheoremsareequivalent.Thenthecontentsofthemareappliedtoexchangetheorder'sconditionofintegral,thelimitandnonpositivefunctionsarealsodiscussed.KeyWords:limittheorem;equivalence;application目录摘要································································································(Ⅰ)Abstract·····························································································(Ⅰ)1引言·······························································································（1）2预备知识·························································································（2）2.1截断函数······················································································（2）2.2函数列两种收敛定义····································································（2）2.3函数L可积定义·············································································（3）3三大极限定理的等价关系研究·····························································（4）3.1Fatou引理的证明········································································（4）3.2用Fatou引理证明列维定理···························································（5）3.3用列维定理证明勒贝格控制收敛定理··············································（6）3.4用勒贝格控制收敛定理证明Fatou引理·············································（8）4三大极限定理的应用········································································（10）4.1在积分与极限交换问题中的应用···················································（10）4.2在非正函数中三大极限定理的应用···············································（13）4.3在推导勒贝格积分逐项积分定理中的应用······································（15）4.4在判断极限函数的可积性中的应用···············································（16）结束语·····························································································（18）参考文献··························································································（19）致谢·································································································（20）11引言勒贝格积分理论是《实变函数论》的中心内容,是数学分析中黎曼积分的推广,它无论在理论上还是应用上都比黎曼积分有许多优越之处,勒贝格积分三大极限定理是勒贝格积分极限理论体系的中心内容,包括勒贝格控制收敛定理、列维定理与Fatou引理.这三大极限定理在许多数学计算和推理中都起到了很重要的作用,其中像使得积分和极限交换问题得到了比在黎曼积分范围内的完满解决,这正是勒贝格积分的一大成功之处.因此,研究勒贝格积分三大极限定理的等价关系及应用有很重要的意义.研究现状:勒贝格积分极限理论体系在《实变函数论》课程中占有十分重要的位置,目前许多研究者的论文中提到了三大极限定理的等价关系及应用,并进行了深入的探讨.像程其襄等人在《实变函数与泛函分析基础》[5]一书中系统地讲述了勒贝格积分的理论知识及极限定理的等价证明;玛哈提·胡斯曼在《勒贝格积分极限定理记注》[7]一文中,从积分极限定理的内容出发,对积分极限定理的的条件及应用展开了讨论;刘世伟在《关于三个积分极限定理的等价性》[1]中围绕着三大积分极限定理的等价关系进行证明;王长辉在《实变函数中几个积分极限定理的应用》[8]中给出了几个极限定理在复函数和实函数中应用的例子;姜功建在《Lebesgue积分在数学分析中的应用》[11]中运用极限定理对一些有关积分的等式、不等式和函数性质的证明进行了论述.这些研究都是在教材的基础上扩展了很大的空间,很多程度上提供了多种对三大极限定理的等价证明方法及应用范围.本文主要工作:针对三大积分极限定理的等价关系及应用进行阐述.在本文的第一部分先介绍了截断函数的定义及性质等预备知识;第二部分对三大极限定理的等价关系进行了证明,因为三大极限定理的等价关系体现在其中一个定理通过某种方法先被证明,那么其他两个定理就可以由此定理推出.所以这里我们先证明Fatou引理,然后用Fatou引理证明列维定理,再用列维定理证明列贝格控制收敛定理,最后用勒贝格控制收敛定理证明Fatou引理,通过这一循环过程,即可完成三大极限定理的等价关系证明;第三部分对三大极限定理在积分与极限交换运算中的应用和非正函数中的应用等内容进行了探讨.通过以上内容的讨论,可以对勒贝格积分极限定理有了进一步的理解,并加深了对它的重要性的应用.22预备知识2.1截断函数2.1.1截断函数的定义定义2.1设mE,)(xf是E上的非负函数,对于任意自然数n,令,(),nnfxfxnfxfxnfxn当时当时即min,nfxfxn,则称nfx为函数)(xf的n—截断函数.2.1.2截断函数的性质从截断函数的定义可得到函数的性质[2]:(1)任意自然数n,Nfx都是E上非负有界函数,(可以用n作为界);(2)截断函数列是单调递增函数列,即12nfxfxfx特别有lim,nnfxfxxE.(3)当函数)(xf在E上非负可测时,则它的每个截断函数Nfx(=1,2,n)都是有界可积的,由截断函数的单调不减性,必有12EEEnfxdxfxdxfxdx,于是极限limnEnfxdx总是存在的（可能是）.2.2函数列两种收敛定义定义2.2(函数列依测度收敛定义)[5]设函数)(xfn是可测集qRE上的一列a.e.有限的可测函数,若有E上a.e.有限的可测函数)(xf满足下列关系:对于任意的0有0limffmEnn,则称函数列)(xfn依测度收敛于)(xf.定义2.3（函数列一致收敛定义）[6]设函数列)(xfn和函数)(xf是定义在同一数集D上,若对任给的正数,总存在某一正整数N,使得当Nn时,对一切Dx,都有)()(xfxfn,则称函数列)(xfn在数集D上一致收敛于)(xf.由函数列依测度收敛和函数列一致收敛的定义可知,当函数列一致收敛时有该函数列是依测度收敛.32.3函数L可积定义定义2.4设0)(xf在可测qRE上可测,定义dxxfdxxfnnE)(lim)(,称为)(xf在E上的L积分.定义2.5设)(xf在可测集qRE上可测.如果在定义4.1的意义下的Edxxf)(与Edxxf)(不同时为,则我们称)(xf在E上积分确定,并定义EEEdxxfdxxfdxxf)()()(为)(xf在E上的L积分,特别当此积分有限时称)(xf在E上L可积.(其中)(xf和)(xf都是E上非负函数，分别称为)(xf的正部和负部.)43三大极限定理的等价关系研究勒贝格积分三大极限定理关系密切,有着内在的联系.为了研究三大定理的等价关系,这里我们先证明Fatou引理,然后用Fatou引理证明列维定理,再用列维定理证明勒贝格控制收敛定理,最后用勒贝格控制收敛定理证明Fatou引理.3.1Fatou引理的证明引理3.1[1]设nfx为集合E上的函数列,令+0=infnnkkUxfx,+0=supnnkkVxfx,Ex.则nUx是E上的递增函数列,nVx是E上的递减函数列,并且lim=limnnnnUxfx,lim=limnnnnVxf