### 2021　Exercises in Analysis C I

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Academic unit or major
Mathematics
Instructor(s)
Tonegawa Yoshihiro
Course component(s)
Exercise    (ZOOM)
Day/Period(Room No.)
Tue5-6(H117)  Tue5-6(H102)
Group
-
Course number
ZUA.C306
Credits
2
2021
Offered quarter
1-2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

This course is an exercise session for the lecture course 'Real Analysis I (ZUA.C305)'. The materials for exercise are chosen from that course.

### Student learning outcomes

Students are expected to:
Be familiar with the notion of sigma-algebra and measure.
Be able to explain the reason why given measurable functions are measurable.
Know the reason why elementary property of integration holds and be able to use them freely.
Be able to apply convergence theorems by checking their assumptions correctly.
Be able to explain the outline of basic construction of measures
Be able to explain the difference between Lebesgue integration and Riemann integration.
Be able to apply the theory of Lebesgue integration to problems in calculus.
Be familiar with the notion of functional inequalities in integration and function spaces defined by integration.
Be able to apply the Fubini theorem to calculate multiple integrals and iterated integrals correctly.

### Keywords

sigma-algebra, measurable space, measure, measure space, Lebesgue measure, measurable function, Lebesgue integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, Hopf's extension theorem, outer measure, Caratheodory measurability, Dynkin system theorem, Riemann integral, H\"older's inequality, Minkowski's inequality, Lebesgue space, product measure, Fubini theorem

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Students are given exercise problems related to what is taught in the course "Real Analysis I".

### Course schedule/Required learning

Course schedule Required learning
Class 1 Overview of measure theory and Lebesgue integration Details will be provided during each class session
Class 2 Sigma-algebra Details will be provided during each class session
Class 3 (Countably additive) measure and its basic properties, completeness Details will be provided during each class session
Class 4 Measurable functions Details will be provided during each class session
Class 5 Definition of integral and its basic properties Details will be provided during each class session
Class 6 Convergence theorems (Monotone convergence theorem, Fatou's lemma and dominated convergence theorem ) and examples Details will be provided during each class session
Class 7 Applications of convergence theorems Details will be provided during each class session
Class 8 Extension theorem for measures Details will be provided during each class session
Class 9 Outer measures and construction of measures Details will be provided during each class session
Class 10 Relation between Riemann integral and Lebesgue integral Details will be provided during each class session
Class 11 L^p-spaces and its completeness, fundamental functional inequalities Details will be provided during each class session
Class 12 Product measure and iterated integral Details will be provided during each class session
Class 13 Fubini theorem and its applications Details will be provided during each class session
Class 14 Extension of Fubini theorem Details will be provided during each class session

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

None required.

### Reference books, course materials, etc.

W. Rudin "Real and complex analysis" McGraw-Hill.
H. Royden "Real Analysis"

Problem session.

### Related courses

• ZUA.C305 ： Real Analysis I
• MTH.C305 ： Real Analysis I
• MTH.C306 ： Real Analysis II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.
Strongly recommended to take ZUA.C305: Real Analysis I (if not passed yet) at the same time.