WSA53000 Probability and Mathematical Statistics C

Hiroshima University Syllabus

Japanese

Academic Year 2026Year School/Graduate School Graduate School of Advanced Science and Engineering (Master's Course) Division of Advanced Science and Engineering Mathematics Program

Lecture Code WSA53000 Subject Classification Specialized Education

Subject Name 確率統計基礎講義Ｃ

Subject Name
（Katakana）カクリツトウケイキソコウギシー

Subject Name in
English Probability and Mathematical Statistics C

Instructor OKAMOTO MAMORU

Instructor
(Katakana) オカモト　マモル

Campus Higashi-Hiroshima Semester/Term 1st-Year,  First Semester,  1Term

Days, Periods, and Classrooms (1T) Tues3-4,Fri3-4：SCI C101

Lesson Style Lecture Lesson Style
(More Details) Face-to-face

Lecture by blackboard

Credits 2.0 Class Hours/Week 4 Language of Instruction J : Japanese

Course Level 5 : Graduate Basic

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students

Keywords Signed measures, Absolute continuity, Radon–Nikodym theorem,  Maximal functions, Differentiation theorem, Conditional expectation, Ergodic theory

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students)
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students)

Class Objectives
/Class Outline A signed measure is a completely additive set function taking real values, and can be regarded as a generalization of a measure. In this course, we study the basic properties of signed measures, in particular the Lebesgue decomposition and the Radon–Nikodym theorem. Furthermore, as applications of these results, we also treat the Riesz representation theorem, the Lebesgue differentiation theorem, and conditional expectation.

Class Schedule 1. Signed measures
2. Decomposition of a measure
3. Absolute continuity
4. Radon–Nikodym theorem
5. Lebesgue spaces
6. Riesz representation theorem
7. Maximal functions
8. Lebesgue differentiation theorem
9. Differentiation of measures
10. Absolutely continuous functions
11. Conditional expectation
12. Conditional probability
13. Measure preserving transformations
14. Ergodic theory
15. Application

Text/Reference
Books,etc. G. B. Folland: Real Analysis, Wiley, 2nd ed
R. Durrett: Probability, Cambridge University Press, 5th ed

PC or AV used in
Class,etc. moodle

(More Details)

Learning techniques to be incorporated

Suggestions on
Preparation and
Review For each lesson, make sure you can state definitions, notation, and theorem statements accurately, and that you can explain the flow and main ideas of theorem proofs.

Requirements

Grading Method Report

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message

Other

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2026Year	School/Graduate School	Graduate School of Advanced Science and Engineering (Master's Course) Division of Advanced Science and Engineering Mathematics Program
Lecture Code	WSA53000	Subject Classification	Specialized Education
Subject Name	確率統計基礎講義Ｃ
Subject Name （Katakana）	カクリツトウケイキソコウギシー
Subject Name in English	Probability and Mathematical Statistics C
Instructor	OKAMOTO MAMORU
Instructor (Katakana)	オカモト　マモル
Campus	Higashi-Hiroshima	Semester/Term	1st-Year, First Semester, 1Term
Days, Periods, and Classrooms	(1T) Tues3-4,Fri3-4：SCI C101
Lesson Style	Lecture	Lesson Style (More Details)	Face-to-face
Lecture by blackboard
Credits	2.0	Class Hours/Week	4	Language of Instruction	J : Japanese
Course Level	5 : Graduate Basic
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students
Keywords	Signed measures, Absolute continuity, Radon–Nikodym theorem, Maximal functions, Differentiation theorem, Conditional expectation, Ergodic theory
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)
Class Objectives /Class Outline	A signed measure is a completely additive set function taking real values, and can be regarded as a generalization of a measure. In this course, we study the basic properties of signed measures, in particular the Lebesgue decomposition and the Radon–Nikodym theorem. Furthermore, as applications of these results, we also treat the Riesz representation theorem, the Lebesgue differentiation theorem, and conditional expectation.
Class Schedule	1. Signed measures 2. Decomposition of a measure 3. Absolute continuity 4. Radon–Nikodym theorem 5. Lebesgue spaces 6. Riesz representation theorem 7. Maximal functions 8. Lebesgue differentiation theorem 9. Differentiation of measures 10. Absolutely continuous functions 11. Conditional expectation 12. Conditional probability 13. Measure preserving transformations 14. Ergodic theory 15. Application
Text/Reference Books,etc.	G. B. Folland: Real Analysis, Wiley, 2nd ed R. Durrett: Probability, Cambridge University Press, 5th ed
PC or AV used in Class,etc.	moodle
(More Details)
Learning techniques to be incorporated
Suggestions on Preparation and Review	For each lesson, make sure you can state definitions, notation, and theorem statements accurately, and that you can explain the flow and main ideas of theorem proofs.
Requirements
Grading Method	Report
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message
Other
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

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