2021年实变函数测试题与答案.pdf

实变函数试题一，填空题1.设1,2nAn,1,2nL,则limnnA.2.,a b:,因为存在两个集合之间的一一映射为3.设E是2R中函数1cos,00,0 xyxx的图形上的点所组成的集合，则E，E.4.若集合nER满足EE,则E为集.5.若,是直线上开集G的一个构成区间,则,满足:,.6.设E使闭区间,a b中的全体无理数集,则mE.7.若()nmEfx()0f x,则说()nfx在E上.8.设nER,0nxR,若,则称0 x是E的聚点.9.设()nfx是E上几乎处处有限的可测函数列,()f x是E上几乎处处有限的可测函数,若0,有,则称()nfx在E上依测度收敛于()f x.10.设()()nfxf x,xE,则()nfx的子列()jnfx,使得.精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 1 页，共 29 页二,判断题.正确的证明,错误的举反例.1.若,A B可测,AB且AB,则mAmB.2.设E为点集,PE,则P是E的外点.3.点集11,2,EnLL的闭集.4.任意多个闭集的并集是闭集.5.若nER,满足*m E,则E为无限集合.三,计算证明题1.证明:ABCABACUI2.设M是3R空间中以有理点(即坐标都是有理数)为中心,有理数为半径的球的全体,证明M为可数集.3.设nER,iEB且iB为可测集,1,2iL.根据题意,若有*0,imBEi,证明E是可测集.4.设P是Cantor集,32ln 1,(),0,1xxPf xxxP.求10(L)()f x dx.5.设函数()f x在Cantor集0P中点x上取值为3x,而在0P的余集中长为13n的构成区间上取值为16n,1,2nL,求10()f x dx.精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 2 页，共 29 页文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y8文档编码:CU2X6W7G8P4 HI5U4H7X8S3 ZA5S6B4P9Y86.求极限:13230lim(R)sin1nnxnxdxn x.实变函数试题解答一填空题1.0,2.2.1(,)cos,0(0,)1x yyxyyxU;.3.闭集.4.ba.5.几乎处处收敛于()f x或a.e.收敛于()f x.6.对000,(,)Ux有0Ex.7.()()nfxf xa.e.于E.二判断题1.F.例如,(0,1)A,0,1B,则AB且AB,但1mAmB.2.F.例如,0(0,1),但 0 不是(0,1)的外点.3.F.由于0EE.4.F.例如,在1R中,11,1nFnn,3,4nL 是一系列的闭集,但是3(0,1)nnFU不是闭集.精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 3 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X45.T.因为若E为有界集合,则存在有限区间I,I,使得EI,则*,m Em II于*m E.三,计算证明题.1.证明如下:2.M中任何一个元素可以由球心(,)x y z,半径为r唯一确定,x,y,z跑遍所有的正有理数,r跑遍所有的有理数.因为有理数集于正有理数集为可数集都是可数集,故M为可数集.3.令1iiBBU,则iEBB且B为可测集,于是对于i,都有iBEBE,故*0imBEmBE,令i,得到*0mBE,故BE可测.从而EBBE可测.4.已知0mP,令0,1GP,则13202210130(L)()(L)ln 1(L)(L)()(L)(L)(R)()133PGGPGf x dxxdxx dxf x dxx dxx dxf x dxx.精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 4 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X45.将积分区间0,1分为两两不相交的集合:0P,1G,2GL,其中0P为Cantor集,nG是0P的余集中一切长为13n的构成区间(共有12n个)之并.由L积分的可数可加性,并且注意到题中的00mP,可得6.因为323sin1nxnxn x在0,1上连续,13230(R)sin1nxnxdxn x存在且与13230(L)sin1nxnxdxn x的值相等.易知由于12x在0,1上非负可测，且广义积分1012dxx收敛，则12 x在0,1上(L)可积，由于323limsin01nnxnxn x，0,1x，于是根据勒贝格控制收敛定理，得到11332323001323010lim(R)sinlim(L)sin11limsin100nnnnxnxnxdxnxdxn xn xnxnx dxn xdx.一、判定下列命题正确与否，简明理由(对正确者予以证明，对错误者举处反例)（15 分，每小题3 分）1.非可数的无限集为c 势集2.开集的余集为闭集。3.若 mE=0，则 E为可数集精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 5 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X44.若|f(x)|在 E 上可测，则f(x)在 E 上可测5.若 f(x)在 E上有界可测，则f(x)在 E上可积二、将正确答案填在空格内（共8 分，每小题2 分）1._可数集之并是可数集。2.A.任意多个 B.c势个 C.无穷多个 D 至多可数个3._闭集之并交是闭集。4.A.任意多个 B.有限个 C.无穷多个 D 至多可数个5.可数个开集之交是_ 6.A 开集 B 闭集 C F型集 D G 型集7.若|f|在 E上可积，则 _ 8.A.f在 E上可积 B.f 在 E上可测 C.f 在 E上有界 D.f在 E上几乎处处有限三、叙述有界变差函数定义、Fatou 引理、Lebesgue 控制收敛定理（共 9 分，每小题3 分）。四、证明下列集合等式（共6 分，每小题3 分）：1.S-S=(S-S)2.Efa=Efa-精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 6 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4五、证明：有限个开集之交是开集。举例说明无限个开集之交不一定是开集。（8 分）六、证明：设f(x)，f(x)为可积函数列，f(x)f(x)a.e于 E，且|f|d|f|d，则对任意可测子集eE有|f|d|f|d（7 分）七、计算下列各题：（每小题5 分，共 15 分）1.sin(nx)d=2.设 f(x)=求d=3.设 f(x)=n=2,3,求d=一、判定下列命题正确与否，简明理由(对正确者予以证明，对错误者举处反例)1.非可数的无限集为c 势集，（不正确！如：直线上的所有子集全体不可数，但其势大于c）。2.开集的余集为闭集。（正确！教材已证的定理）。3.若 mE=0，则 E为可数集（不正确！如contorP集外测度为0，但是 C势集）。精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 7 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X44.若|f(x)|在 E 上可测，则f(x)在 E 上可测（不正确！如）5.若 f(x)在 E上有界可测，则f(x)在 E上可积（不正确！如有界可测，但不可积）二、将正确答案填在空格内1 至多可数个可数集之并是可数集。A.任意多个 B.c 势个 C.无穷多个 D 至多可数个2.有限个闭集之并交是闭集。A.任意多个 B.有限个 C.无穷多个 D 至多可数个3.可数个开集之交是 G 型集A开集 B 闭集 C F型集 D G 型集4.若|f|在 E上可积，则 f在 E上几乎处处有限A.f在 E上可积 B.f 在 E上可测 C.f 在 E上有界 D.f在 E上几乎处处有限三、叙述有界变差函数定义、Fatou 引理、Lebesgue 控制收敛定理（见教材，不赘述！）。精品w o r d 学习资料 可编辑资料-精心整理-欢迎下载-第 8 页，共 29 页文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W10 ZC3L1H3B8X4文档编码:CE6M1J4T1X4 HP8N1X5W8W