Spaces of functions: Continuous functions on locally compact spaces, Stone-Weierstrass theorems, Ascoli-Arzela Theorem. Review of Measure theory: Sigma-algebras,measures, construction and properties of the Lebesgue measure, non-measurable sets, measurablefunctions and their properties. Integration: Lebesgue Integration, various limit theorems,comparison with the Riemann Integral, Functions of bounded variation and absolute continuity.Measure spaces: Signed-measures, Radon-Nikodym theorem, Product spaces, Fubini'sthoerem (without proof) and its applications. Lp-spaces: Holder and Minkowski inequalities,completeness, Convolutions, Approximation by smooth functions. Fourier analysis: FourierTransform, Inverse Fourier transform, Plancherel Theorem for Real numbers.