Course Archives    Theoretical Statistics and Mathematics Unit

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

Midterm 17.pdf Semestral 17.pdf Supplementary and Back Paper 17.pdf Top of the page

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