## Details of MA3204 (Spring 2020)

Level: 3 |
Type: Theory |
Credits: 4.0 |

Course Code | Course Name | Instructor(s) |
---|---|---|

MA3204 |
Analysis IV |
Shibananda Biswas |

Syllabus |
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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 |
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Analysis III (MA3101) |

References |
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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 |