## Details of MA3204 (Spring 2015)

 Level: 6 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
MA3204 Analysis IV Satyaki Mazumder

Syllabus

• Introduction : Drawbacks of Riemann integration; measurement of length: introductory remarks.

• Abstract Measures : Algebra, $\sigma$-algebra and Borel $\sigma$-algebra; outer measure, measure, measure space, measurable set and measurable function.

• Abstract Integration Theory : Definition and properties of the Lebesgue integral; basic convergence theorems: monotone convergence theorem, Fatous lemma and dominated convergence theorem.

• Lebesgue Measure : Construction and basic properties of Lebesgue measure; Ulams theorem and non-measurable sets.

• 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: Hlders, inequality, Jensens inequality and Minkowskis inequalities; definition of $L^p$ spaces, completeness, compact sets in $L^p$ spaces, 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.

• Differentiation Theory : Vitali and Besicovitch covering theorems, differentiation of Radon measures, Lebesgue differentiation theorem, Lebesgue points, absolutely continuous functions, fundamental theorem of calculus, change of variable formula.

• Convolution: Definition and basic properties; mollifiers and approximation by smooth functions.

Prerequisite
Analysis I, Analysis II.

References

1. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.

2. G. B. Folland, Real Analysis: Modern Techniques and Their Applications (2nd Edition), Wiley-Interscience, 1999.

3. S. Kantorovitz, Introduction to Modern Analysis, Oxford University Press, 2003.

4. I. K. Rana, An Introduction to Measure and Integration, (2nd Edition), Graduate Studies in Mathematics, American Mathematical Society, 2002.

5. H. L. Royden, Real Analysis, (3rd Edition), Prentice Hall, 1988.

6. W. Rudin, Real and Complex Analysis (3rd Edition), McGraw-Hill, 1987.

#### Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 2 Core
2 IP 4 Not Allowed
3 MR 2 Not Allowed
4 MR 4 Not Allowed
5 MS 6 Core
6 RS 1 Elective
7 RS 2 Not Allowed