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.
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.