Details of MA3204 (Spring 2020)
| Level: 3 | Type: Theory | Credits: 4.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA3204 | Analysis IV | Shibananda Biswas |
| Syllabus |
|---|
| Introduction: Drawbacks of Riemann integration, measurement of length - introductory remarks.
Lebesgue Measure: Construction and basic properties of Lebesgue measure, non-measurable sets. Abstract Measures: Algebra, _-algebra and Borel _-algebra, outer measure, measure, measure space, measurable set and measurable function. Integration Theory: Definition and properties of Lebesgue integral, basic convergence theorems - monotone convergence theorem, Fatou's lemma and dominated convergence theorem. Borel Measure: Regularity properties of Borel measure, Radon measure, Caratheodory's criterion; Continuity properties of measurable functions - Lusin's and Egoroff's theorems. Lp Spaces: Fundamental inequalities - Holder's inequality, Jensen's inequality and Minkowski's inequality, definition of Lp spaces, completeness, approximation by continuous functions. Signed Measure: Total variation measure, absolute continuity, Lebesgue decomposition, Radon-Nikodym theorem and Hahn decomposition theorem. Product Measure: Measurability in product spaces, product measures, Fubini and Fubini-Tonelli theorems. Convolution: Definition and basic properties, mollifiers and approximation by smooth functions. Differentiation Theory: Lebesgue differentiation theorem, Lebesgue points, absolutely continuous functions, fundamental theorem of calculus, change of variable formula. |
| Prerequisite |
|---|
| Analysis III (MA3101) |
| References |
|---|
| 1. Evans, L.C. and Gariepy, R.F., Measure Theory and Fine Properties of Functions, CRC Press.
2. Folland, G.B., Real Analysis: Modern Techniques and Their Applications (2nd Edi-tion), Wiley-Interscience. 3. Kantorovitz, S., Introduction to Modern Analysis, Oxford University Press. 4. Rana, I.K., An Introduction to Measure and Integration, Narosa Publishers. 5. Royden, H.L., Real Analysis, Prentice-Hall. 6. 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 | 6 | Core |
| 7 | RS | 1 | Not Allowed |
| 8 | RS | 2 | Not Allowed |