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| MATL5131 | Theory of Measure and Integration | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Second Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Ali Hikmet DEĞER | | Co-Lecturer | None | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | The aim of the course is to focus on measure theory and the Lebesque integral, as well as their application to various functional analytic aspects. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | will learn applications of measurement and measurable functions; | 2,3 | 1, | | PO - 2 : | compare the Riemann integral with the Lebesque integral; | 2,3 | 1, | | PO - 3 : | will be able to apply this knowledge in different normed spaces. | 2,3 | 1, | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Fundamentals of real analysis, Preliminary information
Topology and continuity
Sigma algebra, Measurement theory, Measurable functions
Convergence theorems
Integrable functions, Riemann and Lebesque integrals
Applications of the Lebesque integral
Normed spaces and Banach spaces
Linear functionals, Lp-spaces
Hilbert spaces, Fourier analysis
Special topics in integration, Signed measurements
Comparison of measurements, Differentiability and integration |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Fundamentals of real analysis, Preliminary information | | | Week 2 | Topology and continuity | | | Week 3 | Sigma algebra, Measurement theory, Measurable functions | | | Week 4 | 1. Applications | | | Week 5 | Convergence theorems | | | Week 6 | Integrable functions, Riemann and Lebesque integrals | | | Week 7 | Applications of the Lebesque integral | | | Week 8 | 2. Applications | | | Week 9 | Midterm exam | | | Week 10 | Normed spaces and Banach spaces | | | Week 11 | Linear functionals, Lp-spaces | | | Week 12 | Hilbert spaces, Fourier analysis | | | Week 13 | Special topics in integration, Signed measurements | | | Week 14 | Comparison of measurements, Differentiability and integration | | | Week 15 | 3. Applications | | | Week 16 | Final exam | | | |
| 1 | Aliprantis, C.D., Burkinshaw, O. 1990; Principles of Real Analysis, Academic Press, San Diego. | | | |
| 1 | Cheng, S. 1990; A Short Course on the Lebesgue Integral and Measure Theory, Perseus Books. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 15/04/2024 | 2 | 30 | | In-term studies (second mid-term exam) | 12 | 15/05/2024 | 1 | 20 | | End-of-term exam | 16 | 03/06/2024 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 4 | 14 | 56 | | Sınıf dışı çalışma | 8 | 14 | 112 | | Arasınav için hazırlık | 5 | 2 | 10 | | Arasınav | 2 | 1 | 2 | | Kısa sınav | 1 | 1 | 1 | | Dönem sonu sınavı için hazırlık | 14 | 3 | 42 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 225 |
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