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 (M-B101(H102))

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

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.