The Lebesgue Integral:Riemann-Stieltjes integral, Measures and measurable sets, measurable functions, the abstract Lebesgue integral. Product measures and Fubini’s theorem. Complex measures and the lebesgue - Radon - Nikodym theorem and its applications. Function Spaces and Banach Spaces: Lpspaces, Abstract Banach Spaces. The conjugate spaces. Abstract Hilbert spaces.