2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Real Analysis I
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Yoshihiro Tonegawa
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-6 Tue
- Class
- -
- Course Code
- MTH.C305
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 1Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
In this course, we deal with basic concepts and properties of measures and integration by means of measures (Lebesgue integral). We first explain basics of sigma-algebra and (countably additive) measure. It includes the study of Lebesgue measures which are one of the most fundamental measures. We next introduce measurable functions, which are candidates of integrand, and Lebesgue integration, and study their elementary properties. Finally, we study convergence theorems.
They would be a basis of integration theory as well as application of Lebesgue integration. This course will be succeeded by "Real Analysis II" in the second quarter.
The theory of measures and integrations was constructed by Lebesgue on the basis of set theory. These concepts are a natural extension of length, area, volume and probability etc. We can naturally handle operations involving infinity (e.g. limit for figures and functions) within the framework of this theory. In this course, we would like to address how the notion of integration is extended by Lebesgue integration and how effective it is in analysis.
Course description and aims
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.
Keywords
sigma-algebra, measurable space, measure, measure space, Lebesgue measure, measurable function, Lebesgue integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Alternation of standard lecture course and problem session.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Overview of measure theory and Lebesgue integration | Details will be provided during each class session |
| Class 2 | Problem session | Details will be provided during each class session |
| Class 3 | Sigma-algebra | Details will be provided during each class session |
| Class 4 | Problem session | Details will be provided during each class session |
| Class 5 | (Countably additive) measure and its basic properties, completeness | Details will be provided during each class session |
| Class 6 | Problem session | Details will be provided during each class session |
| Class 7 | Measurable functions | Details will be provided during each class session |
| Class 8 | Problem session | Details will be provided during each class session |
| Class 9 | Definition of integral and its basic properties | Details will be provided during each class session |
| Class 10 | Problem session | Details will be provided during each class session |
| Class 11 | Convergence theorems (Monotone convergence theorem, Fatou's lemma and dominated convergence theorem ) and examples | Details will be provided during each class session |
| Class 12 | Problem session | Details will be provided during each class session |
| Class 13 | Applications of convergence theorems | Details will be provided during each class session |
| Class 14 | Problem session | Details will be provided during each class session |
Study advice (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.
Textbook(s)
None required.
Reference books, course materials, etc.
Basic of Lebesgue integral, Masanori Hino
Evaluation methods and criteria
Examination, report and presentation.
Related courses
- MTH.C306 ： Real Analysis II
Prerequisites
Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.