Details of MA3203 (Spring 2021)
| Level: 3 | Type: Theory | Credits: 4.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA3203 | Analysis-IV | Shibananda Biswas |
| Syllabus |
|---|
| \it Introduction: Drawbacks of Riemann integration, measurement of length-- introductory remarks.
\it Abstract Measures: Algebra, $\sigma$-algebra and Borel $\sigma$-algebra, outer measure, measure, Caratheodory extension Theorem and construction of Lebesgue measure on $\mathbbR^n$ as an application, measure space, measurable set and measurable function. \it Integration Theory: Definition and properties of Lebesgue integral, basic convergence theorems-- monotone convergence theorem, Fatou's lemma and dominated convergence theorem. \it Borel Measure: Regularity properties of Borel measure, Radon measure, Caratheodory's criterion; Continuity properties of measurable functions-- Lusin's and Egoroff's theorems. \it $L^p$ Spaces: Fundamental inequalities - H\"older's inequality, Jensen's inequality and Minkowski's inequality, definition of $L^p$ spaces, completeness, approximation by continuous functions. \it Product Measure: Measurability in product spaces, product measures, Fubini and Fubini-Tonelli theorems, polar coordinates and change of variable theorem.\\ |
| Prerequisite |
|---|
| Analysis III (MA3101) |
| References |
|---|
| 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 | Core |
| 2 | IP | 4 | Not Allowed |
| 3 | IP | 6 | Not Allowed |
| 4 | MR | 2 | Not Allowed |
| 5 | MR | 4 | Not Allowed |
| 6 | MS | 10 | Not Allowed |
| 7 | MS | 4 | Not Allowed |
| 8 | MS | 6 | Core |
| 9 | MS | 8 | Elective |
| 10 | RS | 1 | Not Allowed |
| 11 | RS | 2 | Not Allowed |