Classes of sets and measures: Sigma algebras, Borel sigma algebra, measure and its properties (monotonicity, continuity etc.), Carathéodory's extension theorem and construction of Lebesgue measure on the real line.
Integration: Lebesgue integration, Monotone, Dominated convergence theorems, Fatou’s lemma, modes of convergence, Egoroff’s and Lusin’s theorems, Lp space - definition and examples.
Product spaces, product σ-algebras and measures, Lebesgue measure on R^n , the Fubini and Tonelli theorems, change of variable.
Absolute continuity, Radon-Nikodym theorem, Lebesgue decomposition, signed and complex measures, Hahn-Jordan decomposition theorems, Riesz representation theorem for C(K).
Differentiation and integration - functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus for Lebesgue integrals.
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- Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics Vol 126, American Mathematical Society, 2011
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- Frank Jones, Lebesgue Integration On Euclidean Space, Jones and Bartlett Publishers, Inc