Lebesgue measure on Rⁿ : Introduction, outer measure, measurable sets, Lebesgue measure, regularity properties, a nonmeasurable set, measurable functions, Egoroff's theorem, Lusin's theorem. Lebesgue integration: Simple functions, Lebesgue integral of a bounded function over a set of finite measure, bounded convergence theorem, integral of nonnegative functions, Fatou's Lemma, monotone convergence theorem, the general Lebesgue integral Lebesgue convergence theorem, change of variableformula. Differentiation and integration: Functions of bounded variations, differentiation of an integral, absolutely continuity, L^p-spaces: The Minkowski's inequality and Holder's inequality, completeness of L^p, denseness results in L^p; Fourier series: Definition of Fourier series, formulation of convergence problems, L² theory of Fourier series, convergence of Fourier series.