Details of MA3208 (Spring 2023)
| Level: 3 | Type: Theory | Credits: 4.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA3208 | Measures and Integrals | Shirshendu Chowdhury |
| Syllabus |
|---|
| Introduction: Drawbacks of Riemann integration, measurement of length introductory
remarks. Abstract Measures: Algebra, alpha-algebra and Borel alpha-algebra, outer measure, measure, Caratheodory extension Theorem and construction of Lebesgue measure on R n as an application, measure space, measurable set and measurable function. Integration Theory: Definition and properties of Lebesgue integral, basic convergence theorems monotone convergence theorem, Fatous lemma and dominated convergence theorem. Borel Measure: Regularity properties of Borel measure, Radon measure, Caratheodorys criterion; Continuity properties of measurable functions Lusins and Egoroffs theorems. L p Spaces: Fundamental inequalities - Holders inequality, Jensens inequality and Minkowskis inequality, definition of L p spaces, completeness, approximation by continuous functions. Product Measure: Measurability in product spaces, product measures, Fubini and FubiniTonelli theorems, polar coordinates and change of variable theorem. |
| References |
|---|
| Suggested Texts:
1. De Barra, G., Measure Theory and Integration, New Age International Publishers. 2. Evans, L.C. and Gariepy, R.F., Measure Theory and Fine Properties of Functions, CRC Press. 3. Folland, G.B., Real Analysis: Modern Techniques and Their Applications (2nd Edition), Wiley-Interscience. 4. Kantorovitz, S., Introduction to Modern Analysis, Oxford University Press. 5. Rana, I.K., An Introduction to Measure and Integration, Narosa Publishers. 6. Royden, H.L., Real Analysis, Prentice-Hall. 7. Rudin, W., Real and Complex Analysis, McGraw-Hill. |
Course Credit Options
| Sl. No. | Programme | Semester No | Course Choice |
|---|---|---|---|
| 1 | IP | 2 | Not Allowed |
| 2 | IP | 4 | Not Allowed |
| 3 | IP | 6 | Not Allowed |
| 4 | MP | 2 | Not Allowed |
| 5 | MP | 4 | Not Allowed |
| 6 | MR | 2 | Not Allowed |
| 7 | MR | 4 | Not Allowed |
| 8 | MS | 10 | Not Allowed |
| 9 | MS | 4 | Not Allowed |
| 10 | MS | 6 | Not Allowed |
| 11 | MS | 8 | Not Allowed |
| 12 | RS | 1 | Not Allowed |
| 13 | RS | 2 | Not Allowed |