MA6110 Topics in Advanced Analysis (modified)
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; Lp spaces on a general measure space, and its completeness; Vitali's convergence theorem; Dual of Lp for 1 ≤ p < ∞; Chebychev's inequality. (10 Lectures)
Unit-3: Positive linear functionals on Cc(X), where X is a locally compact Hausdorff space; Riesz representation theorem; Luzin's theorem; Vitali Caratheodory theorem; Denseness of Cc(X) in Lp(X) for 1 ≤ p < ∞; Dual of C0(X), Riesz-Thorin interpolation theorem. (15 Lectures)
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.
1. H.L. Royden, Real Analysis, Third edition, Prentice-Hall of India, 1995.
2. S. Kesavan, Measure and Integration, TRIM, 2019.