Prerequisite: Advanced and multi-variable calculus. This course is an introduction to measure and integration, the theory of metric spaces and their applications to the approximation of real valued functions. It starts with notions of convergence from sequences of continuous functions, the Ascoli- Arzela compactness theorem and the Weierstrass approximation theorem. The main body of the course deals with the theory of measure and integration and limiting processes for the Lebesgue integral. The last part covers the topics of differentiation, functions of bounded variation and Fourier series. The course provides the main background needed in modern Advanced Mathematics related to Real Analysis.