課程資訊

Real Analysis (Ⅰ)

107-1

MATH7201

221 U2870

3.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1071MATH7201_W

Midterm Exam 30%, Final Exam 30% and Homework 40%.

Office Hours

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.

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.

(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