**MTH ****404: Measure and Integration**** ****(4)**

**Pre-requisites**: MTH 403 Real Analysis II

Topology of the real line, Borel, Hausdorff and Lebesgue measures on the real line, regularity properties, Cantor function

σ-algebras, measure spaces, measurable functions, integrability, Fatou’s lemma, Lebesgue’s monotone convergence theorem, Lebesgue’s dominated convergence theorem, Egoroff’s theorem, Lusin’s theorem, the dual space of C(**X**) for a compact, Hausdorff space, **X **

Comparison with Riemann integral, improper integrals

Lebesgue’s theorem on differentiation of monotonic functions, functions of bounded variation, absolute continuity, differentiation of the integral, Vitali’s covering lemma, fundamental theorem of calculus

Holder’s, inequality, Minkowski’s inequality, convex functions, Jensen’s inequality, **L**^{p} spaces, Riesz-Fischer theorem, dual of **L**^{p} spaces

*Suggested Books*:

- W. Rudin,
*Real and Complex Analysis*, third edition. Tata-McGraw Hill, 1987 - H. Royden,
*Real Analysis*, third edition, Prentice-Hall of India, 2008 - R. Wheeden, A. Zygmund,
*Measure and Integral*, Taylor and Francis, 1977 - J. Kelley, T. Srinivasan,
*Measure and Integral*, Volume I, Springer, 1987 - Rana,
*An Introduction to Measures and Integration*, Narosa Publishing House - E. Lieb, M. Loss,
*Analysis*, Narosa Publishing House

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