Course Archives Theoretical Statistics and Mathematics Unit |
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Course: Analysis I Time: Currently not offered Level: Ph. D First year | |||

Syllabus Past Exams Syllabus 1. Measure Theory: Sigma-algebras, measures, outer measures, completion, construction and properties of the Lebesgue measure, non-measurable sets, Measurable functions, point wise convergence, almost uniform convergence, convergence in measure. 2. Integration: Lebesgue integration, limit theorems, comparison with the Riemann integral, relationship with differentiation, functions of bounded variation and absolute continuity. 3. Signed Measures: Radon - Nikodym theorem, Lebesgue decomposition theorem, change of variable formula, Product Spaces, Fubini's theorem and applications. 4. Lp-Spaces : Holder and Minkowski inequalities, completeness, convolutions, approximation by smooth functions, duality. 5. Riesz representation theorem: Riesz representation theorem for positive linear functionals, Proof of the theorem, construction of the Lebesgue measure via this approach. Top of the page Past Exams | |||

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