Real analysis.pdf

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1、实变函数笔记cnyouwei@gmail.com⃝c2011-2012前言i前言献给亲爱的钟老师.欢迎大家mailto:cnyouwei@gmail.com批评指正!持续更新，请关注http://iask.sina.com.cn/u/1776601564/ish.γ.ω.2012年5月于南京大学目录前言i概论10.1连续函数的极限..........................10.2曲线长度..............................10.3微分与积分....................

2、.........20.4Riemann积分的局限性......................30.5Lebesgue积分与Lebesgue测度.................4第一章集合与点集61.1集合的运算.............................61.2映射与基数.............................101.2.1映射.............................101.2.2特征函数与幂集......................111.2.3基数.

3、............................131.3Rn中的点集............................201.3.1Rn中的度量........................201.3.2领域,内点,开集,闭集...................211.3.3极限点,闭包........................231.3.4Cantor集..........................251.3.5闭集上连续函数的延拓定理...............301.4Bo

4、rel集与纲性定理........................34第二章Lebesgue可测集382.1环上的测度.............................382.1.1环Rn............................3802.1.2环Rn上的测度......................4202.2外测度................................482.3Lebesgue测度...........................532.4Lebesgue

5、可测集与Borel集的关系...............582.5Lebesgue测度的完备性问题...................622.5.1Lebesgue不可测集的存在性...............62ii目录iii2.5.2测度的延拓.........................62第三章Lebesgue可测函数673.1Lebesgue可测函数的定义及其性质...............673.2Lebesgue可测函数的结构.....................693.3几乎处处收敛与

6、依测度收敛...................753.3.1几乎处处收敛........................753.3.2依测度收敛.........................783.4函数可测的充要条件及复合函数的可测性...........84第四章Lebesgue积分874.1测度有限集上非负可测函数的积分...............874.2一般可测集上一般可测函数的积分...............944.3Lebesgue积分的极限定理.....................1

7、064.4Lebesgue积分与连续函数之间的关系..............1164.5Fubini定理.............................118第五章测度导数与Newton-Lebniz公式1235.1Vitali覆盖定理...........................1235.2Hahn分解定理...........................1285.3Radon测度的导数.........................1335.4Radon-Nikodym定理.....

8、..................1405.5单调函数与有界变差函数.....................1455.5.1单调函数..........................1455.5.2有界变差函数........................1495.6绝对连续函数............................157参考文献162索引163