Home for Real Analysis I (Fall)

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.

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.