Exercises in Analysis C I

Academic unit or major

Mathematics

Instructor(s)

Yoshihiro Tonegawa

Class Format

Exercise (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

5-6 Tue

Class

-

Course Code

ZUA.C306

Number of credits

020

Course offered

2024

Offered quarter

1-2Q

Syllabus updated

Mar 14, 2025

Language

Japanese

Syllabus

Course overview and goals

This course is an exercise session for the lecture course 'Real Analysis I (ZUA.C305)'. The materials for exercise are chosen from that course.

Course description and aims

Students are expected to:
Be familiar with the notion of sigma-algebra and measure.
Be able to explain the reason why given measurable functions are measurable.
Know the reason why elementary property of integration holds and be able to use them freely.
Be able to apply convergence theorems by checking their assumptions correctly.
Be able to explain the outline of basic construction of measures
Be able to explain the difference between Lebesgue integration and Riemann integration.
Be able to apply the theory of Lebesgue integration to problems in calculus.
Be familiar with the notion of functional inequalities in integration and function spaces defined by integration.
Be able to apply the Fubini theorem to calculate multiple integrals and iterated integrals correctly.

Keywords

sigma-algebra, measurable space, measure, measure space, Lebesgue measure, measurable function, Lebesgue integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, Hopf's extension theorem, outer measure, Caratheodory measurability, Dynkin system theorem, Riemann integral, H\"older's inequality, Minkowski's inequality, Lebesgue space, product measure, Fubini theorem

Competencies

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

Class flow

Students are given exercise problems related to what is taught in the course "Real Analysis I".

Course schedule/Objectives

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)

None required.

Reference books, course materials, etc.

Basic of Lebesgue integral, Masanori Hino

Evaluation methods and criteria

presentation.

Related courses

• ZUA.C305 ： Real Analysis I
• MTH.C305 ： Real Analysis I
• MTH.C306 ： Real Analysis II

Prerequisites

Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.
Strongly recommended to take ZUA.C305: Real Analysis I (if not passed yet) at the same time.