2022 | Additive and nonadditive measure theories

2022　Additive and nonadditive measure theories

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Academic unit or major: Graduate major in Mathematical and Computing Science

Instructor(s): Murofushi Toshiaki

Class Format: Lecture (Face-to-face)

Media-enhanced courses

Day/Period(Room No.): Mon5-6(W832) Thr5-6(W832)

Group: -

Course number: MCS.T420

Credits: 2

Academic year: 2022

Offered quarter: 4Q

Syllabus updated: 2022/12/19

Lecture notes updated: -

Language used: English

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Syllabus

Course description and aims

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.

Student learning outcomes

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 that will be developed

✔ Specialist skills

Intercultural skills

Communication skills

Critical thinking skills

Practical and/or problem-solving skills

Class flow

Lectures and exercises

Course schedule/Required learning

	Course schedule	Required learning
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.

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.

Textbook(s)

Not specified.

Reference books, course materials, etc.

References are provided in the lectures.

Assessment criteria and methods

Will be based on exercise and/or report.

Related courses

MCS.T201 ： Set and Topology I
MCS.T304 ： Lebesgue Interation

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

Not specified.

Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).

Toshiaki MUROFUSI (murofusi[at]c.titech.ac.jp)

Office hours

Specified by the lecturer.

Other

To be announced in lectures.

TOKYO INSTITUTE OF TECHNOLOGY