Real Analysis II

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Yoshihiro Tonegawa

Class Format

Lecture/Exercise (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-6 Tue

Class

-

Course Code

MTH.C306

Number of credits

110

Course offered

2025

Offered quarter

2Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

This course is a continuation of "Real Analysis I" in the first quarter. In this course, we deal with more advanced concepts and properties of measures and integration by means of measures (Lebesgue integration). We first explain construction and extension of measure. Second, we show the relation between Lebesgue integral and Riemann integral. Third, we introduce function spaces defined by means of integration and studies their basic properties. Finally, we study the Fubini theorem as a measure-theoretic treatment of (iterated) integral on product spaces.
The theory of measures and integrations was constructed by Lebesgue on the basis of set theory. These concepts are a natural extension of length, area, volume and probability etc. We can naturally handle operations involving infinity (e.g. limit for figures and functions) within the framework of this theory. In this course, we would like to address how the notion of integration is extended by Lebesgue integration and how effective it is in analysis.

Course description and aims

Students are expected to:
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

Hopf's extension theorem, outer measure, Caratheodory measurability, Dynkin system therem, 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

Alternation of standard lecture course and problem session.

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

Examination, report and presentation.

Related courses

• MTH.C305 ： Real Analysis I
• MTH.C201 ： Introduction to Analysis I

Prerequisites

Student are required to have passed Real Analysis I.
Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.