課程資訊
課程名稱
 實分析一
Real Analysis (Ⅰ) 
開課學期
 107-1 
授課對象
 理學院  數學研究所  
授課教師
 王振男 
課號
 MATH7201 
課程識別碼
 221 U2870 
班次
  
學分
 3.0 
全/半年
 半年 
必/選修
 必修 
上課時間
 星期一3,4(10:20~12:10)星期三3,4(10:20~12:10) 
上課地點
 天數102天數102 
備註
 總人數上限：50人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1071MATH7201_W 
課程簡介影片
 
核心能力關聯
 核心能力與課程規劃關聯圖
課程大綱
為確保您我的權利,請尊重智慧財產權及不得非法影印
課程概述

這門一學年的課是實分析的入門課程，主要是提供數學系所，應數所，及外系需要嚴格分析工具的學生修習。實分析是進階分析理論的基礎，也是其他領域常用的重要工具，例如調和分析，偏微分方程，機率論等等。第一學期的課將著重在基本的Lebesgue測度理論，了解這個理論，對於進入更抽象的測度理論有很大的幫助。我們將從較具體的外測度來引進Lebesgue測度，切入可測函數及Lebesgue的積分理論。Lebesgue積分的優點在於強大的收斂的定理，這些都是分析問題常用的工具，熟悉這些操作對於往後的學習相當重要。有了Lebesgue的積分理論，我們就可以介紹L^p空間甚至更抽象的函數空間，L^2空間有許多很好的性質，基本的傅立葉分析及建立在此空間。我們也將會介紹Lebesgue微分理論，這是屬於調和分析的入門材料。除此，我們也會介紹Radon-Nikodym 定理及其他測度的理論。  

課程目標
 提供學生實分析的理論基礎，課程將著重在具體的測度論的建立，使學生能具有最基本的觀念及熟悉分析工具的操作。 
課程要求
 Advanced Calculus

Midterm Exam 30%, Final Exam 30% and Homework 40%. 
預期每週課後學習時數
  
Office Hours
 每週二 10:00~12:00 
指定閱讀
 1. Measure and Integral: An Introduction to Real Analysis by Richard L. Wheeden, Antoni Zygmund, Second Edition (Chapman & Hall/CRC Pure and Applied Mathematics).
2. Real Analysis by H.L. Royden, Third Edition.
 
參考書目
 1. Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis) (Bk. 3), First Edition by Elias M. Stein, Rami Shakarchi.
2. Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland, Second Edition.
3. A User-Friendly Introduction to Lebesgue Measure and Integration by Gail S. Nelson, AMS.
4. Real Analysis and Probability (Cambridge Studies in Advanced Mathematics) by R. M. Dudley, Second Edition. 
評量方式
(僅供參考)
  
No.
項目
百分比
說明
1. 
 Homework  
40% 
  
2. 
 Midterm 
30% 
  
3. 
 Final 
30% 
  
 
課程進度
週次
日期
單元主題
第1週
 9/10,9/12  Motivations, Riemann integrals, Lebesgue's idea, Lebesgue outer measure, Basic properties of Lebesgue outer measure. 
第2週
 9/17,9/19  Rotational invariant of the outer measure, Lebesgue measurable sets, Examples of measurable sets (open sets, intervals, closed sets, etc.), sigma-algebra, the smallest sigma-algebra, Borel sigma-algebra, Countable union of disjoint measurable sets. 
第3週
 9/24,9/26  Quiz session 
第4週
 10/01,10/03  Continuity of the measure, Characterization of a measurable set, Caratheodory characterization, Dynkin's pi-\lambda lemma, Measurable sets under Lipschitz transforms, Construction of a non-measurable set 
第5週
 10/08,10/10  Construction of a non-measurable set, Measurable functions, Borel measurable functions, Composition of measurable functions. 
第6週
 10/15,10/17  Properties of measurable functions, Characteristic functions, Simple functions, Convergence to measurable functions, Semicontinuous functions and measurable functions, Egorov's theorem, Lusin's theorem, Convergence in measure, Pointwise convergence implies convergence in measure. 
第7週
 10/22,10/24  Cauchy criterion of convergence in measure, Lebesgue integral, Area under the graph of a measurable function, Monotone convergence theorem, Lebesgue integration in Riemann's idea, Tchebyshev's inequality, Fatou's lemma, Lebesgue's dominated convergence theorem, The integral of a measurable function, Integrability. 
第8週
 10/29,10/31  Lebesgue integral of an arbitrary measurable function, Lebesgue integrable functions, Monotone convergence theorem, Fatou's lemma, Lebesgue's dominated convergence theorem, Lebesgue's bounded convergence theorem, Lebesgue integrability vs Riemann integrability, Distribution function, Riemann-Stieltjes integral.
10/31, quiz session 
第9週
 11/05,11/07  11/7, midterm 
第10週
 11/12,11/14  Self-study week 
第11週
 11/19,11/21  Iterated integrals, Fubini's theorem, Tonelli's theorem. 
第12週
 11/26,11/28  Convolution of functions, Marcinkiewicz theorem, Indefinite integrals, Set-valued functions, Continuity, Absolute continuity, Lebesgue's differentiation theorem, Hardy-Littlewood maximal function, Hardy-Littlewood maximal operator, Weakly integrable functions.  
第13週
 12/03,12/05  Mapping property of Hardy-Littlewood maximal function in L^p, Points of density, Points of dispersion, Lebesgue's points and Lebesgue's set, Family of sets shrinking regularly, Vitali's covering lemma.  
第14週
 12/10,12/12  Covering in the Vitali sense, Vitali covering lemma, Monotone increasing functions, Derivatives, Fundamental theorem of Calculus. 
第15週
 12/17,12/19  Functions of bounded variation, Derivative of the total variation, Fubini's lemma, Absolutely continuous functions, Singular functions, Necessary and sufficient conditions of an absolutely continuous function, Decomposition of a function of bounded variation, Integration by parts. 
第16週
 12/24,12/26  Convex functions, Jensen's inequality, Characterization of convexity, Change of variables, Differentiable mappings, Jacobian matrix, Critical values, Sard's theorem, Diffeomorphsim.  
第17週
 12/31,1/02  Quiz sessions