MA747 Measure Theory L-T-P-C [4-0-0-8]
Fields and a-fields, generators; Borel a-field on Euclidean, metric and general topological spaces. Monotone classes, monotone class theorem. Finitely additive measures. Measures, finite and a-finite measures. Borel measures, regularity. Outer measures; Caratheodory's extension theorem. Lebesgue measure in Euclidean spaces. Distribution functions. Measurable functions and their properties. Induced measures. a-fields generated by classes of functions; Monotone class theorem for functions. Integrability of functions. Lebesgue integrals and their properties. Fatou's lemma, monotone convergence theorem, dominated convergence theorem. Finite-dimensional product measurable spaces and measures on them. Product measures. Fubini's Theorem. Holder's. Minkowski's and Jensen 's inequalities. LP spaces; Characterizations of compact, precompact sets in LP spaces. Complex-valued measurable and integrable functions. Fourier transforms of finite measures on the real line and inversion formulae. Signed and complex-valued measures. Absolute continuity and singularity of measures. Lebesgue's differentiation theorem. Hahn decomposition theorem. Radon-Nikodym Theorem. Lebesgue decomposition theorem. Spaces of measures. Weak convergence. Helly's Theorem. Measures on locally compact spaces. Radon measures; Riesz representation theorem.
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