Home for Real Analysis I (Fall)
開版大吉，歡迎蒞臨，課程簡介。
上課內容摘要
Week 1
1. Explain the incompleteness of the Riemann integral.
2. Define a measure on all subsets of any given set X with certain requirements is not possible.
3. Define a ring, an algebra, and a σ-algebra.
4. Characterize the Borel σ-algebra of the real line (useful on many occasions).
5. Basic limiting properties of a measure.
6. Expect homework next week.
Week 2
1. Homework #1 (due 9/29). Solutions
2. Discuss the Caratheodory construction of measures.
3. Uniqueness of the measure extended from a countably additive nonnegative set function defined on a ring or an algebra.
Week 3
1. Sorry, no break! Homework #2 (due 10/6). Solutions
2. Due to typhoon, HW#1 is due on 10/1.
3. Unique extension of a measure using Dynkin's π-λ theorem.
4. Discussion of semirings and rings. We will define Borel and Lebesgue measures on the real line.
Week 4
1. Homework #3 (due 10/13) Solutions
2. Discussion the completion of a measure.
Week 5
1. Homework #4 (due 10/20) Solutions
2. Regularity of Lebesgue measure.
3. The existence of a nonmeasurable set.
4. Definition of a measurable function.
Week 6
1. Homework #5 (due 10/27) Solutions
*由於上課進度改變，第1,2題習題併入下禮拜作業。課堂給的問題，只對那個課堂構造的不可測集。
2. Product σ-algebra and Borel σ-algebra of the product topology.
3. If f,g are measurable, then f+g and f·g are measurable.
Week 7
1. Homework #6 (due 11/3)
2. Definition of integral for a measurable function.
3. For any nonnegative measurable function, there exist a sequence of nonnegative simple functions converge to it monotonically.
4. Definition of integrable functions.
5. Note on nonmeasurable sets.
Week 8
1. 因下週為期中考週，本週作業暫停一次。
2. Three important convergence theorems: monotone convergence theorem, Fatou's lemma, bounded convergence theorem.
3. Relations between different concepts of convergence.
4. Some counterexamples.
5. Homework #7 (due 11/19)
Week 9
1. Monotone class theorem.
2. Existence of the product measure.
3. Tonelli-Fubini's theorem.
4. Tonelli-Fubini's theorem for complete measure.
5. n-dimensional Lebesgue measure.
6. Homework #8 (due 11/24)
Week 10
No class on 11/17. Quiz section on 11/19.
Week 11
1. Tonelli-Fubini's theorem for complete measures.
2. The n-dimensional Lebesgue measure.
3. Signed measures, Hahn decomposition, Jordan decomposition.
4. Homework #9 (due 12/8) (check your email).
Week 12
1. Absolute continuity.
2. The Lebesgue-Radon-Nikodym Theorem.
3. The Radon-Nikodym derivative.
4. Differentiation with respect to the Lebesgue measure.
5. Homework #10 (due 12/15).
Week 13
1. Covering lemma.
2. The Hardy-Littlewood Maximal Theorem.
3. The Lebesgue Differentiation Theorem.
4. The Differentiation Theorem for a regular Radon measure.
5. Homework #11 (due 12/22).
Week 14
1. Vitali's covering lemma and its application.
2. Application of the Lebesgue Differentiation Theorem.
3. Functions of bounded variation.
4. Correspondence of an NBV function and a complex Borel measure.
5. No homework this week.