Existence of Lebesgue non-measurable set. Lebesgue Differentiation: Hardy-Littlewood maximal function, density points, Radon-Nikodym theoremįunctional Analysis: Banach and Hilbert spaces, dual spaces, L p-spaces. Integration: convergence theorems, Fubini's theorem, convolutions Measure Theory: abstract measures and σ-algebras, Monotone Class Theorem, outer measures and Caratheorodry's theorem, Borel sets and Lebesgue measure, signed measures Lieb and Michael Loss, Analysis AMS Graduate Texts in Mathematics, 14 (either edition) Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces.Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley 2nd edition, 1999 Teaching Assistant: Abhishek Oswal Hours: Fri 2:00 - 3:00 PM.
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