实分析、测度与积分
Point set topology of Rn
Countable and uncountable sets, the axiom of choice, Zorn’s lemma. Metric spaces. Completeness; separability; compactness; Baire category; uniform continuity; connectedness; continuous mappings of compact spaces. Functions on topological spaces. Equicontinuity and Ascoli’s theorem; the Stone-Weierstrass theorem; topologies on function spaces; compactness in function spaces.
Measure and integration
Measures; Borel sets and contor sets; Lebesgue measures; distributions; product measures. Measurable functions. approximation by simple functions; convergence in measure; Construction and properties of the integral; convergence theorems; Radon-Nykodym theorem; Fubini’s theorem; mean convergence. Monotone functions; functions of bounded variation and Borel measures; Absolute continuity, convex functions; semicontinuity.
1， 超限归纳法、递归原理、势、选择公理、集列的上极限、下极限与极限。
1。集合系（半环、环、代数、sigma-代数等），这些系统的不同性质。
2， 集代数、Sigma-代数、集类生成的Sigma-代数、可测空间、Borel集、集环、集半环、Sigma-环、Borel Sigma-代数、可加测度、可数可加测度、
2。半环上的测度，sigma-可加空间上半环的Lebesgue经典测度。
3。从半环到最小环的测度的连续性，Lebesgue和Jordan外测度，Lebesgue和Jordan测度，他们的性质。
测度、测度的完备性与连续性，Borel测度、直线上的Lebesgue-Stieltjes测度，概率测度、概率空间、可数可加性的判据、紧类、逼近类、具有逼近紧类的测度的可数可加性、Lebesgue测度。
3， 外测度、mu-可测集、测度的完备化、测度的Lebesgue扩张、无限测度、Sigma-有限测度。可测集结构的理论。
4， R^n上的Lebesgue测度与Lebesgue可测集、Jordan可测集、Lebesgue—Stieltjes 测度、集合的单调类、集合的Sigma-可加类、单调类定理、Suslin集、Suslin运算、Suslin集。
5， Caratheodory外测度、正则外测度、任意Borel集m-可测的充要条件。
6， 可测函数、他们的性质，可测函数及其极限。可测空间、Borel可测、可测函数的基本性质、处处与几乎处处收敛性、他们的性质。
Egoroff定理、Cauchy函数列、Riesz定理、Luszin 定理、简单
函数的Lebesgue积分及其性质。
7，一般情形下Lebesgue积分的一般定义、Lebesgue积分的基本性质、
10。Lebesgue积分号下取极限。
11。Lebesgue积分的一致连续，可测集上可积性的Lebesgue准则，Chebyshev不等式、具有无限测度的空间上的积分。
8， Lebesgue可积函数空间的完备性、Lebesgue控制收敛定理、Levi单调收敛定理、Fatou定理、可积性的判据。
9，（区间上）Riemann积分与Lebesgue积分的关系、变量替换，符号测度、符号测度的Hahn分解与Jordan分解、Radon-Nikodym定理、测度空间的乘积。
10，测度的直和，Fubini定理、测度的无穷乘积、测度在映射下的像、适合Luszin性质的映射、R^n上的变量替换。
11， Holder与Minkowski不等式、L^p空间、Lp空间的完备性、L^p空间上的逼近。
18。Lebesgue积分的微分。
19。绝对连续函数及其与Lebesgue积分的关系.
20。Lebesgue积分的变量替换与分部积分。
21。Hilbert空间、Cauchy-Buniakowsky公式。
22。Hilbert空间上的展开定理。
23。正交系与Hilbert空间的基。
24。Hilbert-Schmidt正交化过程。
12， 作为Hilbert空间的L^2空间、L^2空间上的正交基、Bessel不等式、Parseval等式。Riesz-Fisher定理、Chebyshev-Hermite多项式、实直线上函数的微分、上下导数。Hilbert空间上的线性泛函。
13， 有界变差函数、绝对连续函数、不定积分的绝对连续性、绝对连续性与不定积分的关系、Newton-Lerbniz公式、绝对连续函数的分部积分公式、Vitali覆盖定理。
Real Analysis & Measure and Integration
- Royden, Real Analysis, except chapters 8, 13, 15.
- E.M. Stein and R. Shakarchi; Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005
- 周民强, 实变函数论, 北京大学出版社, 2001
- 夏道行等，《实变函数论与泛函分析》，人民教育出版社.
Rudin, “Real & Complex Analysis”
Rudin, “Functional Analysis”
1，A.N.Kolmogorov、S.V.Fomin，函数论与泛函分析初步，物理数学书籍出版社，2004。
2，I.P.Natanson，实变函数论，科学出版社，1974。
3，V.I.Bogachev，测度论基础，“正则与混沌动力学”出版社，2006。
4。M.I.Dyachenko、P.L.Ulyanov，测度与积分，法克特里亚出版社，2002。
《实变函数论习题集》捷利亚科夫斯基
Halmos，”Measure Theory”(GTM 18)
E.Hewitt, K.Stromberg “Real and Abstract Analysis”(GTM 25)
Folland, Real analysis：
J.Oxtoby Measure and Category(GTM2)
Donald L. Cohn, “Measure Theory”
陈建功 “实函数论”
鄂强《实变函数的例题与习题》, 《实变函数论的定理与习题》高等教育出版社
徐森林《实变函数论》中国科学技术大学出版社
郑维行《实变函数与泛函分析概要》（第一册）高等教育出版社
《实变函数》江泽坚，吴志泉
严加安，《测度论讲义》，科学版
程士宏，《测度论与概率论》，北大版
习题
程民德,邓东皋 “实分析”
那汤松 “实变函数论”
汪林 “实分析中的反例”
“实变函数论习题解答”
“实变函数论的定理与习题”
【习题集与辅导书】
156《实变函数与泛函分析:定理•方法•问题》胡适耕，刘金山编著
158《实变函数内容、方法与技巧》孙清华，孙昊著
Real analysis 、 Measure and integration
Point set topology of Rn
Countable and uncountable sets, the axiom of choice, Zorn’s lemma. Metric spaces. Completeness; separability; compactness; Baire category; uniform continuity; connectedness; continuous mappings of compact spaces. Functions on topological spaces. Equicontinuity and Ascoli’s theorem; the Stone-Weierstrass theorem; topologies on function spaces; compactness in function spaces.
Measure and integration
Measures; Borel sets and contor sets; Lebesgue measures; distributions; product measures. Measurable functions. approximation by simple functions; convergence in measure; Construction and properties of the integral; convergence theorems; Radon-Nykodym theorem; Fubini’s theorem; mean convergence. Monotone functions; functions of bounded variation and Borel measures; Absolute continuity, convex functions; semicontinuity.
1 , Principle of transfinite induction, recursion, the potential, the axiom of choice, set the upper limit and lower limit and limit.
1 。 Collection of (semi-ring, ring, algebra,Sigma- algebra), the different nature of these systems.
2 , Set algebra,Sigma- algebra, the set class generated Sigma- algebras and measurable spaces,Borel set, a set of rings, set half rings,Sigma- rings, Borel Sigma- algebra, measure, countably additive measure, can be added
2 。 Semiring measuringSigma- half ring on the space of Lebesgue classical measure.
3 。 From the smallest rings half ring measuring continuity ofLebesgue and Jordan outer measure,Lebesgue and Jordan measure their properties.
Measure, Completeness and continuity of the measure, Borel Measure, On the line Lebesgue-Stieltjes Measure, Probability, probability spaces, countable additivity criterion, compact type, approximation, a close measure of the compact class of countable additivity, and Lebesgue Measure.
3 , Outer measure,Mu- measurement of complete set, measure, measure of the Lebesgue expansion, infinite measure andSigma- finite measure. Structure of measurable set theory.
4 , R^n on the Lebesgue measure and Lebesgue measurable sets,Jordan measurable sets,Lebesgue-Stieltjes Monotone class of the measurement, collection, collection of Sigma- plus class, monotone class theorem,Suslin sets,Suslin operation,SuslinSet.
5 , Caratheodory outer measure and outer measure, arbitrary Borel set m- measurable if and only if.
6 , Measurable functions, their properties, measurable functions and limits. Measurable spaces,Borel measurable, observable function of basic properties,and almost everywhere convergence, and their properties.
Egoroff Theorem, Cauchy Function column, Riesz Theorem, Luszin Theorem, simple
Function Lebesgue Integral and its properties.
7 , Under normal circumstances Lebesgue General definition of integral, Lebesgue Basic properties of integral,
10 。 Lebesgue integration under the limit.
11 。 Lebesgue integral is uniformly continuous and integrable on a measurable set of Lebesgue criterionChebyshev inequality, integral with infinite measure space.
8 , Lebesgue integrable functions space completeness, andLebesgue control convergence theorem,Levi the monotone convergence theorem,Fatou theorem, the criterion for integrability.
9 ,( On the interval) Riemann Integral and Lebesgue Integral Relationships, Variable substitution, Signed measure, signed measuresHahn Decomposition and Jordan Decomposition, Radon-Nikodym Theorem, the product of a measure space.
10 , Measure straight and, Fubini Theorem, the infinite product measure, measure map, suitable for Luszin Property mapping,R^n On the variable substitution.
11 , Holder and the Minkowski inequality,L^p space,Lp space is complete, andL^p approximation of the space.
18 。 Lebesgue integral of the differential.
19 。 Absolute continuous function and its Lebesgue integral relations.
20 。 Lebesgue integral variable substitution and integration by parts.
21 。 Hilbert space,Cauchy-Buniakowsky formula.
22 。 Hilbert space expansion theorem.
23 。 Orthogonal and Hilbert space base.
24 。 Hilbert-Schmidt orthogonalization process.
12 , As the Hilbert space L^2 spaceL^2 space on the orthogonal basis,Bessel inequality, Parseval equality. Riesz-Fishertheorems,Chebyshev-Hermite polynomial, and the differential, and the derivative of a function on the real line. Hilbertspace of linear functionals.
13 , Function of bounded variation and absolute continuous functions, indefinite integral of absolute continuity and absolute continuity and the indefinite integral relation,Newton-Lerbniz formula, the absolutely continuous functions integration by parts formula, theVitali covering theorem.
Real Analysis & Measure and Integration
1. Royden, Real Analysis, except chapters 8, 13, 15.
2. E.M. Stein and R. Shakarchi; Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005
3. Zhou Min strong , Theory of functions of real variables , Peking University Press , 2001
4. Xia Daoxing, of the real variable function theory and functional analysis, the people’s education press .
Rudin, ” Real & Complex Analysis “
Rudin, ” Functional Analysis “
1 , A.N.Kolmogorov 、 S.V.Fomin , Theory of functions and functional analysis, physics and mathematics Books Publishing House, 2004 。
2 , I.P.Natanson , Theory of functions of a real variable, science press, 1974 。
3 , V.I.Bogachev And measures on the basis of ” Regular and chaotic dynamics ” Publishing House, 2006 。
4 。 M. I. Dyachenko, andp. L. Ulyanov, measure and integral, faketeliya Publishing House,2002.
Of the sets of real variable function theory of the Czech liyakefusiji
Halmos ,”Measure Theory”(GTM 18)
E.Hewitt, K.Stromberg “Real and Abstract Analysis”(GTM 25)
Folland, Real analysis :
J.Oxtoby Measure and Category(GTM2)
Donald L. Cohn, “Measure Theory”
Chen jiangong ” Theory of real functions”
Jaw strong examples and exercises of the functions of real variables , The real variable function theory theorems and exercises of higher education press
Xu Senlin of the real variable function theory University of science and technology of China press
Zheng Weihang an outline of the real analysis and functional analysis (book) by higher education press
Jiang Zejian of the real variable function, Wu Zhiquan
Strictly, lectures on the measure theory, science
CHENG Shihong, measure theory and probability theory, North Edition
Exercises
Cheng teh- , Deng donggao ” Real analysis”
The Tang Song ” Theory of functions of real variables”
Wang Lin ” In real analysis example”
“Theory of functions of real variables questions and problems”
“Real variable function theory theorems and exercises”
“The problem sets and books”
156 The real analysis and functional analysis : Theorem • method • problems of Hu Shi Geng and Liu Jinshan authoring
158 Of the functions of real variable contents, methods and techniques of Qinghua Sun, Sun h a
Serendipities 