課程資訊

Real Analysis (Ⅰ)

111-1

MATH7201

221 U2870

3.0

The first semester of this course will cover the following topics :
1. Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
2. Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem,
Fubini’s theorem
3. Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem,
separation principles and Hahn-Banach theorem, Hilbert spaces

The topics in the second semester will include:
4. Differentiation and Integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation,
absolutely continuous functions, differentiability of functions
5. L^p spaces
6. Abstract Measure and Signed Measures: absolute continuity, Radon-Nikodym Theorem
7. Convolution operators and Fourier Transform

Course Goal：This course aims to introduce basic theory and techniques of modern analysis.

Introduction to Mathematical Analysis I, II; Linear Algebra

Office Hours

(1) Fon-Che Liu, Real Analysis, Oxford University Press (2) Elias M. Stein and Rami Shakarchi, Real Analysis

(僅供參考)

 No. 項目 百分比 說明 1. Homework 30% 2. Midterm Exam 30% 3. Final Exam 40%

 課程進度
 週次 日期 單元主題 第1週 9/5-9/7 0. Introduction: Background and motivation: (0) continuous nowhere differentiable functions, functions of bounded variation, Cantor's set, a space-filling curve, Borel's measurable sets, Lebesgue's theory, non-measurable sets (1) Length, area, volume (2) Fourier series (3) Limits of Integrals 1. Measure theory 1-1 Properties of area (length, volume) 1-2 Rectangles and open sets 第2週 9/12-9/14 1. Measure theory 1-3 Jordan exterior (outer) measure 1-4 Lebesgue exterior (outer) measure Examples, Cantor set, Equivalent definitions of the Lebesgue exterior measure Properties of exterior measure: monotonicity, sub-additivity 第3週 9/19-9/21 1-4 Lebesgue exterior (outer) measure Properties of exterior measure: monotonicity, sub-additivity, open sets approximation, disjoint sets with a distance 1-5 Measurable sets and the Lebesgue measure Motivation Basic properties: (1) Open sets, measure zero sets, and closed sets are measurable. (2) A countable union of measurable sets is measurable. 第4週 9/26-9/28 1-5 Measurable sets and the Lebesgue measure (3) The complement of a measurable is measurable (4) A countable intersection of measurable sets is measurable. Theorem: Countable additivity holds for disjoint measurable sets Sigma-algebra 第5週 10/3-10/5 1-5 Measurable sets and the Lebesgue measure - Increasing and decreasing sequences of sets - sigama-algebra and Borel sets: G_delta sets and F_sigma sets, relation between Borel sets and Lebesgue measurable sets - Non-measurable sets 第6週 10/10-10/12 - Invariance properties: translation, reflection, dilation - Caratheodory measurable 第7週 10/17-10/19 1-6 Measurable function - Riemann integral and Lebesgue integral - Definition: equivalent definitions - Continuous functions are measurable - sup, inf , limsup, liminf, and lim of measurable functions - sum and product of measurable functions 第8週 10/24-10/26 - measure zero - almost everywhere 1-7 Approximation by simple functions and step functions - approximation by simple function 第9週 10/31-11/1 - approximation by step functions - midterm examination 第10週 11/7-11/9 1-8 Littlewood's three principles - Egorov's Theroem - Lusin's Theorem - Littlewood's three principles Chapter 2 Integration Theorey 2-1 The Lebesgue integral - 4 steps: simple function, bounded functions supported on a set of finite measure, non-negative functions, general functions 第11週 11/14-11/16 - Step 1 simple functions: Independence of representation, linearity, additivity, monotonicity, triangle inequality - Step 2 bounded functions supported on a set of finite measure: limit of the integrals of convergent simple functions 第12週 11/21-11/23 - Step 2 bounded functions supported on a set of finite measure: linearity, additivity, monotonicity, triangle inequality, Bounded Convergent Theorem, Riemann integrable implies Lebesgue integrable - Step 3 nonnegative functions: linearity, additivity, monotonicity, 第13週 11/28-11/30 - Step 3 Fatou's lemma, Monotone Convergence Theorem - Step 4 general case: integrable functions, linearity, additivity, monotonicity, Dominated Convergence Theorem, invariance properties 2-2 L1 space: completeness of L1 第14週 12/5-12/7 2-2 L1 space: convergence in L1 implies convergence of a subsequence almost everywhere; approximation by simple functions, step functions, or continuous functions of compact support; invariance properties; complex valued functions 第15週 12/12-12/14 2-3 Fubini's Theorem: statement and proof