 Introduction : Drawbacks of Riemann integration; measurement of length: introductory remarks.
 Abstract Measures : Algebra, $\sigma$algebra and Borel $\sigma$algebra; outer measure, measure, measure space, measurable set and measurable function.
 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 nonmeasurable sets.
 Borel Measure : Regularity properties of Borel measure, Radon measure, Caratheodorys criterion; continuity properties of measurable functions : Lusins and Egoroffs theorems.
 $L^p$ Spaces : Fundamental inequalities: Hlders, inequality, Jensens inequality and Minkowskis inequalities; definition of $L^p$ spaces, completeness, compact sets in $L^p$ spaces, approximation by continuous functions.
 Signed Measure : Total variation measure, absolute continuity, Lebesgue decomposition, RadonNikodym theorem and Hahn decomposition theorem.
 Product Measure : Measurability in product spaces, product measures, Fubini and FubiniTonelli 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.
