Details of MA3204 (Spring 2020)

Level: 3 Type: Theory Credits: 4.0

Course CodeCourse NameInstructor(s)
MA3204 Analysis IV Shibananda Biswas

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.

Analysis III (MA3101)

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.ProgrammeSemester NoCourse 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