MSM1101 - Measure Theory

HNGU M.SC. MATHEMATICS SEMESTER - I

MSM1101 - MEASURE THEORY

Revision: Standard topology on R, structure of open sets, cantor set, lim sup, lim inf.

Unit 1: 

• Algebra and σ-algebra of sets, 
• σ-algebra of Boral sets, 
• Lebesgue outer measure on R,
• Measurable sets, 
• Lebesgue measure.

Unit 2: 

• Measurable function, 
• Littelwood’s three principles, 
• Egoroff’s theorem, Integral of a simple function, 
• Lebesgue integral of bounded functions, 
• Bounded convergence theorem.

Unit 3: 

• Integral of nonnegative functions, 
• General Lebesgue (integral), 
• Fatou’s lemma,
• Monotone convergence theorem,
• Lebesgue’s convergence theorem,
• Convergence in measure.

Unit 4: 

• Differentiation of monotone functions,
• Functions of bounded variation,
• Differentiation of an integral,
• Absolutely continuous functions and indefinite integrals.

Download Reference Books:

The course is covered by Real Analysis - H. L. Ryoden, Macmillan Pub. Co. 3rd Ed.

1. Theory of Functions of a Real Variable – I. N. Natansen, Fredrik Pub. Co., 1964.

2. Measure Theory – P. R. Halmos, East and West Press.

3. Introduction to Real Variable Theory – S. C. saxena and S. N. shah Prentice Hall of India, 1980.

4. Real and Complex Analysis – Rudin, W, 3nd Edition, Tata McGraw-Hill Publishing Co. Ltd., 1974.

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