Μeasure Theory
| Title | ΘΕΩΡΙΑ ΜΕΤΡΟΥ ΚΑΙ ΟΛΟΚΛΗΡΩΣΗΣ / Μeasure Theory |
| Code | 0643 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter |
| Coordinator | Romanos diogenis Malikiosis |
| Common | Yes |
| Status | Active |
| Course ID | 40000031 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2018-sīmera)
Registered students: 7
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Core Courses A2 | 1 | 1 | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Winter |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600290415
|
Type Of Offer
- Disciplinary Course
Course Type 2021
Specialization / Direction
Course Type 2011-2015
General Foundation
Mode of Delivery
- Face to face
- Distance learning
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600290415
- eLearning: https://elearning.auth.gr/course/view.php?id=8303
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
Prerequisites
General Prerequisites
Background on Real Analysis of undergraduate level (sequences and series of real numbers, continuous functions, derivatives, Riemann integral, sequences and series of functions).
Learning Outcomes
Upon successful completion of the course, students will:
1. know the basic theory of Lebesgue Measure and Integration
2. be acquainted with modern definitions and notions of Mathematical Analysis
3. establish the required foundations for other Analysis courses in the graduate program
4. acquire experience in the study of international literature in English
5. be able to study high level contemporary research articles in the area of Mathematical Analysis
General Competences
- Retrieve, analyse and synthesise data and information, with the use of necessary technologies
- Adapt to new situations
- Make decisions
- Work autonomously
- Work in teams
- Generate new research ideas
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
Outer Lebesgue measure, measurable sets, measurable functions, integrable functions. Fubini's theorem. Modes of convergence. Hardy-Littlewood inequality, Lebesgue differentiation theorem and application to convolution kernels. Differentiability of increasing functions, functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus. Abstract measure theory, Carathéodory's theorem, absolutely continuous and singular measures, Radon-Nikodym theorem.
Keywords
Lebesgue measure, integral, Fubini's theorem, Hardy-Littlewoord inequality, Lebesgue differentiation theorem, absolutely continuous functions, functions of bounded variation, absolutely continuous measures, singular measures, Radon-Nikodym theorem
Educational Material Types
- Notes
- Slide presentations
- Multimedia
- Book
Use of Information and Communication Technologies
Use of ICT
- Use of ICT in Communication with Students
- Use of ICT in Student Assessment
Description
Online exercise hours.
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | 1.3 | |||
| Reading Assigment | 183 | 6.1 | |||
| Tutorial | 13 | 0.4 | |||
| Project | 20 | 0.7 | |||
| Written assigments | 40 | 1.3 | |||
| Exams | 5 | 0.2 | |||
| Total | 300 | 10 |
Student Assessment
Description
The final grade for this course will be calculated as follows:
1. Exam: 80%
2. Written Projects: 20%
In the case that a midterm exam takes place in addition to the final exam, the midterm and final exam grades have the same weight. Specifically, each of them corresponds to 40% of the final grade.
Student Assessment methods
- Written Exam with Short Answer Questions (Formative, Summative)
- Written Exam with Extended Answer Questions (Formative, Summative)
- Written Assignment (Formative, Summative)
- Performance / Staging (Formative, Summative)
Bibliography
Course Bibliography (Eudoxus)
Δ. Μπετσάκος, Εισαγωγή στην Πραγματική Ανάλυση, Εκδόσεις Αφοί Κυριακίδη, 2016.
Additional bibliography for study
1. E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.
2. G. B. Folland, Real Analysis: Modern Techniques and Applications, 2nd edition, Wiley Interscience, 1999.
3. T. Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, Volume 126, American Mathematical Society, 2011.
4. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, 2nd edition, Chapman & Hall/CRC Pure and Applied Mathematics, Routledge, 2015.
5. W. Rudin, Real and Complex Analysis, 3rd edition, Higher Mathematics Series, McGraw Hill, 1986.
Last Update
08-05-2025
