2021 Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science

Lebesgue Interation

Academic unit or major: Undergraduate major in Mathematical and Computing Science
Instructor(s): Hideyuki Miura / Jin Takahashi / Shinya Nishibata / Toshiaki Murofushi / Shunsuke Tsuchioka / Shunsuke Ichiki
Class Format: Lecture/Exercise
Media-enhanced courses: -
Day of week/Period (Classrooms): 5-6 Mon (W833) / 5-8 Thu (W833)
Class: -
Course Code: MCS.T304
Number of credits: 210
Course offered: 2021
Offered quarter: 1Q
Syllabus updated: Jul 10, 2025
Language: Japanese

Syllabus

Course overview and goals

We give an introduction on measures and Lebesgue integration, which are notions refining and extending area and integration. After introducing Lebesgue (outer) measure and measurable functions in the 1-dimensional Euclidean space, we define the Lebesgue integral and explain the fundamental properties.
We present convergence theorems concerning exchange between limit and integration for sequences of functions.
Next we extend the theory of Lebesgue integration to multi dimensional Euclidean spaces and present Fubini's theorem
on the relation between the multiple integral and the iterated integral.
Finally, we deal with the Lebesgue spaces and related inequalities for functions.

Course description and aims

Understand the fundamentals on the theory of measures and Lebesgue integration. Apply convergence theorems and Fubini's theorem to specific problems.

Keywords

Outer measure, measurable sets, measure, measurable functions, Lebesgue integral, convergence theorems, Fubini's theorem

Competencies

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

Class flow

For understanding of this course, it is necessary to be skilled at the contents by the calculation by hand. Therefore the exercise class is given every two weeks.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Overview on the theory of Lebesgue integral	Understand the contents of the lecture.
Class 2	Lebesgue outer measure	Understand the contents of the lecture.
Class 3	Exercise for preliminaries and Lebesgue outer measure	Cultivate a better understanding of lectures.
Class 4	Lebesgue measurable set	Understand the contents of the lecture.
Class 5	Definition on Lebesgue measurable functions	Understand the contents of the lecture.
Class 6	Exercise for Lebesgue measurable set and Lebesgue measurable functions	Cultivate a better understanding of lectures.
Class 7	Properties on Lebesgue measurable functions	Understand the contents of the lecture.
Class 8	Definition of Lebesgue integral	Understand the contents of the lecture.
Class 9	Exercise for Lebesgue measurable functions and Lebesgue integral	Cultivate a better understanding of lectures.
Class 10	Properties of Lebesgue integral	Understand the contents of the lecture.
Class 11	Convergence theorems	Understand the contents of the lecture.
Class 12	Exercise for Lebesgue integral and convergence theorems	Cultivate a better understanding of lectures.
Class 13	Convergence theorems	Understand the contents of the lecture.
Class 14	Relation between Lebesgue integral and Riemann integral	Understand the contents of the lecture.
Class 15	Exercise for convergence theorems and relation between Lebesgue integral and Riemann integral	Cultivate a better understanding of lectures.
Class 16	Product measure	Understand the contents of the lecture.
Class 17	Fubini's theorem	Understand the contents of the lecture.
Class 18	Exercise for product measure and Fubini's theorem	Cultivate a better understanding of lectures.
Class 19	フビニの定理の応用	Understand the contents of the lecture.
Class 20	Lebesgue spaces	Understand the contents of the lecture.
Class 21	Exercise for Fubini’s thoerem and Lebesgue spaces	Cultivate a better understanding of lectures.

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

To be announced

Reference books, course materials, etc.

To be announced.

Evaluation methods and criteria

By scores of the examination and the reports.

Related courses

MCS.T301 ： Vector and Functional analysis

Prerequisites

The students are encouraged to understand the limit, the supremum (infimum), and the fundamentals on sets and topology.