Ph.D Mathematics Fall / Spring Semester

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MP-660 Advanced Measure Theory

Overview

Riemann-Stieltjes and Lebesgue integration. Classical Banach Spaces. Weierstrass’ approximation theorem. Riemann-Stieltjes integration. Lebesgue measurable sets. Lebesgue measure. Lebesgue measurable functions. Lebesgue integral functions. Properties of the Lebesgue integral. Fubini’s theorem. Absolutely continuous functions. Differentiation under the integral sign. Classical Banach Spaces. LP –spaces. Convergence and completeness in LP – spaces. Bounded linear functional on the LP –spaces. General convergence theorem. Singed measures. The Radon-Nikodym theorem. Product measures. Inner measure. Extension by sets of measure zero. Caratheodory outer measure. Hausdorff measure


Credit hours/ Marks:- (3 credits)

1. Royden, H.L.: Real Analysis (3rd edition) ( Macmillan, New York, 1988). 2. Saks, S.: Theory of the Integrals Vol. VII (Hafner Publishing Company, 1937). 3. Halmos, P.R.: Measure Theory (Von Nostrand, New York, 1950). 4. Bartle, R.G.: The Elements of Integration and Lebesgue Measure (Wiley Classics Library, 1995). 5. Loomis, L. H.: An Introduction to Abstract Harmonic Analysis (Van Nostrand, 1953). 6. Riesz, F. and Nagay.B.S.: Functional Analysis (Ungar Publishing Co., New York, 1955).

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