Measure Theory and Integration (MA40042)

Semester I, 2020-21

Content:

Systems of measurable sets: sigma-algebras, pi-systems, d-systems, Dynkin's Lemma, Borel sigma-algebras. Measure in the abstract: convergence properties, Uniqueness Lemma, Caratheodory's Theorem (statement). Lebesgue outer measure and measure on Rn. Measurable functions. Monotone-Class Theorem. Probability. Random variables. Independence. Integration of non-negative and signed functions. Monotone-Convergence Theorem. Fatou's Lemma. Dominated-Convergence Theorem. Expectation. Product measures. Tonelli's and Fubini's Theorem. Radon-Nikodym Theorem (statement). Inequalities of Jensen, Holder, Minkowski. Completeness of Lp.

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A mock online paper