### MA6110 Topics in Advanced Analysis (modified)

#### Course Details

Unit-1: Review of general positive measure and integral; Signed measures; Hahn decomposition and Jordan decomposition; Lebesgue Radon-Nykodym theorem; Complex measures and Radon-Nykodym theorem for complex measures; Total variation norm. (15 Lectures)}

Unit-2: Stone Weiestrass theorem; L^{p} spaces on a general measure space, and its completeness; Vitali's convergence theorem; Dual of L^{p} for 1 ≤ p < ∞; Chebychev's inequality. (10 Lectures)

Unit-3: Positive linear functionals on C_{c}(X), where X is a locally compact Hausdorff space; Riesz representation theorem; Luzin's theorem; Vitali Caratheodory theorem; Denseness of C_{c}(X) in L^{p}(X) for 1 ≤ p < ∞; Dual of C_{0}(X), Riesz-Thorin interpolation theorem. (15 Lectures)

#### Course References:

Text Books:

1. W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, International Editions, 1987.

2. G. Folland}, Real Analysis: Modern techniques and Their Applications, Second Edition,
John Wiley & Sons, INC, 1999.

Reference Books:

1. H.L. Royden, Real Analysis, Third edition, Prentice-Hall of India, 1995.

2. S. Kesavan, Measure and Integration, TRIM, 2019.