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Syllabus Introduction : Drawbacks of Riemann integration; measurement of length: introductory remarks. Abstract Integration Theory : Definition and properties of the Lebesgue integral; basic convergence theorems: monotone convergence theorem, Fatous lemma and dominated convergence theorem.
Lebesgue Measure : Construction and basic properties of Lebesgue measure; Ulams theorem and non-measurable sets. Borel Measure : Regularity properties of Borel measure, Radon measure, Caratheodorys criterion; continuity properties of measurable functions : Lusins and Egoroffs theorems. Signed Measure : Total variation measure, absolute continuity, Lebesgue decomposition, Radon-Nikodym theorem and Hahn decomposition theorem. Product Measure : Measurability in product spaces, product measures, Fubini and Fubini-Tonelli theorems.
Differentiation Theory : Vitali and Besicovitch covering theorems, differentiation of Radon measures, Lebesgue differentiation theorem, Lebesgue points, absolutely continuous functions, fundamental theorem of calculus, change of variable formula. Convolution: Definition and basic properties; mollifiers and approximation by smooth functions. References L. Evans and R. Course Credit Options.
EVANS GARIEPY MEASURE THEORY PDF
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