MTH404A - Analysis II

Lebesgue measure on Rn: Introduction, outer measure, measurable sets, Lebesgue measure, regularity properties, a non measurable set, measurable functions, Egoroffs theorem, Lusins theorem. Lebesgue integration: Simple functions, Lebesgue integral of a bounded function over a set of finite measure, bounded convergence theorem, integral of nonnegative functions, Fatuous Lemma, monotone convergence theorem, the general Lebesgue integral, Lebesgue convergence theorem, change of variable formula. Differentiation and integration: Functions of bounded variation, differentiation of an integral, absolutely continuity, Lpspaces: The Minkowskis inequality and Hlders inequality, completeness of Lp, denseness results in Lp. Fourier series: Definition of Fourier series, formulation of convergence problems, The L2 theory of Fourier series, convergence of Fourier series.