The course aims to cover the Lebesgue theory of measure and integration as well as differentiation theorem of Lebesgue. Their role in the analysis of functions of real variables is emphasized. The fact that the real number system is linearly ordered is duly brought to light, in that monotony results are stressed. Topics covered include the following: Metric Spaces, Semicontinuities; Measurable space and Measure space, Measurable function and integration, Egorov theorem, Monotone convergence theorem, Fatou lemma and Lebesgue dominated convergence theorem; Construction of measures through outer measures according to Caratheodory, Regularities of outer measures and measure-theoretical approximation of sets; L^p spaces and their dual spaces with related inequalities; Covering lemmas and Lebesgue differentiation theorem for Radon measure and indefinite integral; Differentiation of functions of a real variable, Functions of bounded variation and absolutely continuous functions, Change of variables formula for integrals in Euclidean space; Basic principles of linear analysis and Hilbert space.