The first semester of this course will cover the following topics :
1. Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
2. Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem,
Fubini’s theorem
3. Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem,
separation principles and Hahn-Banach theorem, Hilbert spaces

The topics in the second semester will include:
4. Differentiation and Integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation,
absolutely continuous functions, differentiability of functions
5. L^p spaces
6. Abstract Measure and Signed Measures: absolute continuity, Radon-Nikodym Theorem
7. Convolution operators and Fourier Transform