


関連するディプロマポリシー Related Diploma Policy

時代の課題と社会の要請に応えた専門的知識と技能/Expert knowledge and skills to address the issues of the age and the demands of society



到達目標 Target to be Reached

After the course, the students are expected to be able to:
understand the basic concept of measure theory,
describe the construction of Lebesgue integral,
use the famous three convergence theorems and Fubini's theorem,
understand the basic structure of L^p spaces and RadonNikodym theorem.


授業内容 Course Content

Theory of Lebesgue integral is crucial to understand modern analysis (functional analysis, probability theory etc.).
The pourpose of this lecture is to introduce its basic theory.
The course covers the following topics:
abstract measure and integration theory,
the Lebesgue measure and the Lebesgue integral on the Euclidean space,
convergence theorems, Fubini's theorem, L^p spaces and RadonNikodym theorem.


授業計画 Course Planning

01. Introduction
02. Measure theory 1 (sigma algebras and Borel sets)
03. Measure theory 2 (basic properties)
04. Construction of Lebesgue integral
05. Basic properties of Lebesgue integral
06. Monotone convergence Theorem
07. Lebesgue dominated convergence theorem
08. Fubini's theorem
09. L^p spaces 1 (Hoelder's inequality and Minkowski's inequality)
10. L^p spaces 2 (completeness)
11. Riesz representation theorem
12. RadonNikodym theorem 1 (absolute continuity)
13. RadonNikodym theorem 2 (proof)
14. Summary
Homework will be given in ``almost every" lecture.
Your solutions to the assigned homework will be graded and returned to you.


授業運営 Course Management

Lecture


評価方法 Evaluation Method

The final grade will be based on the homework.


オフィスアワー Office Hour (s)

The students may ask questions by email.



参考書 Book (s) for Reference

W. Rudin,Real and Complex Analysis,3rd edition,McGrawHill Book Co.
荷見守助『関数解析入門』第２版[内田老鶴圃]


