2023 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Additive and nonadditive measure theories
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Toshiaki Murofushi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon (W8E-306(W832)) / 5-6 Thu (W8E-306(W832))
- Class
- -
- Course Code
- MCS.T420
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 4Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The classical (additive) measure theory provides a background for study in both functional analysis and probability theory.
The first aim of this course is to help students acquire an understanding of the basics of the classical measure theory.
The second aim is to learn the idea of a non-additive extension of the classical measure theory.
Course description and aims
The first goal of this course is to master the basics of the classical (additive) measure theory, and the second goal is to understand the basic concepts in the non-additive measure theory.
Keywords
measures, the Lebesgue integral, non-additive measure, the Choquet integral
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Lectures and exercises
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Measurable spaces | Understand the contents covered by the lecture. |
| Class 2 | Definition and properties of measure | Understand the contents covered by the lecture. |
| Class 3 | Construction of measures | Understand the contents covered by the lecture. |
| Class 4 | Lebesgue measure spaces | Understand the contents covered by the lecture. |
| Class 5 | Measurable functions | Understand the contents covered by the lecture. |
| Class 6 | Definition of integral | Understand the contents covered by the lecture. |
| Class 7 | Properties of integral | Understand the contents covered by the lecture. |
| Class 8 | Convergence theorems | Understand the contents covered by the lecture. |
| Class 9 | Function spaces | Understand the contents covered by the lecture. |
| Class 10 | Convergence concepts | Understand the contents covered by the lecture. |
| Class 11 | Product measures and Fubini's theorem | Understand the contents covered by the lecture. |
| Class 12 | Signed measures | Understand the contents covered by the lecture. |
| Class 13 | Radon-Nikodym's theorem | Understand the contents covered by the lecture. |
| Class 14 | Non-additive measures and the Choquet integral | Understand the contents covered by the lecture. |
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)
Not specified.
Reference books, course materials, etc.
Umegaki Hisaharu, Ohya Masanori, Tsukada Makoto ``Measures, Integral, and Probabilities'' (in Japanese) Kyoritsu Publ.
Sugeno Michio, Murofushi Toshiaki ``Fuzzy Measures'' (in Japanese) Nikkan Kogyo Shinbun
Evaluation methods and criteria
Will be based on exercise and/or report.
Related courses
- MCS.T201 ： Set and Topology I
- MCS.T304 ： Lebesgue Interation
Prerequisites
Not specified.
Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).
MUROFUSHI Toshiaki (murofusi[at]c.titech.ac.jp)
Office hours
Specified by the lecturer.
Other
To be announced in lectures.