Real Analysis I

Academic unit or major

Mathematics

Instructor(s)

Yoshihiro Tonegawa

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Tue

Class

-

Course Code

ZUA.C305

Number of credits

200

Course offered

2024

Offered quarter

1-2Q

Syllabus updated

Mar 14, 2025

Language

Japanese

Syllabus

Course overview and goals

In this course, we deal with concepts and properties of measures and integration by means of measures (Lebesgue integral). We first explain basics of sigma-algebra and (countably additive) measure. It includes the study of Lebesgue measures which are one of the most fundamental measures. We next introduce measurable functions, which are candidates of integrand, and Lebesgue integration, and study their elementary properties. We next study convergence theorems. They would be a basis of integration theory as well as application of Lebesgue integration. We next explain construction and extension of measure. We next show the relation between Lebesgue integral and Riemann integral. Then, 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. We strongly recommend to take this course with "Exercises in Analysis C I".
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 familiar with the notion of sigma-algebra and measure.
Be able to explain the reason why given measurable functions are measurable.
Get to 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

Standard lecture course.

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.C306 ： Real Analysis II
• ZUA.C306 ： Exercises in Analysis C I

Prerequisites

Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.
Strongly recommended to take ZUA.C306: Exercises in Analysis C I (if not passed yet) at the same time.