2020 Additive and nonadditive measure theories

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Academic unit or major
Graduate major in Mathematical and Computing Science
Instructor(s)
Murofushi Toshiaki 
Course component(s)
Lecture     
Day/Period(Room No.)
Mon5-6(Zoom)  Thr5-6(Zoom)  
Group
-
Course number
MCS.T420
Credits
2
Academic year
2020
Offered quarter
4Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

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 and Radon-Nykodym's theorem Understand the contents covered by the lecture.
Class 13 Non-additive measures Understand the contents covered by the lecture.
Class 14 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

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

Not specified.

Other

To be announced in lectures.

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