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Indian Institute of Science Education and Research Berhampur

Mathematics

MTH 404: Measure and Integration (4)

Pre-requisites: MTH 403 Real Analysis II

Topology of the real line, Borel, Hausdorff and Lebesgue measures on the real line, regularity properties, Cantor function

σ-algebras, measure spaces, measurable functions, integrability, Fatou’s lemma, Lebesgue’s monotone convergence theorem, Lebesgue’s dominated convergence theorem, Egoroff’s theorem, Lusin’s theorem, the dual space of C(X) for a compact, Hausdorff space, X

Comparison with Riemann integral, improper integrals

Lebesgue’s theorem on differentiation of monotonic functions, functions of bounded variation, absolute continuity, differentiation of the integral, Vitali’s covering lemma, fundamental theorem of calculus

Holder’s, inequality, Minkowski’s inequality, convex functions, Jensen’s inequality, Lp spaces, Riesz-Fischer theorem, dual of Lp spaces

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