The first semester of this course will cover the following topics : set theory, functions of bounded variation, Riemann-Stieltjes integrals, Lebesgue measure theory, measurable functions, Lebesgue integration theory, differentiation theory and their applications. The topics in the second semester include function spaces with focus on the classical Lp spaces, convolution operators, abstract measures and some basic materials of Harmonic analysis such as Fourier series.