This is a course aiming at graduate students in mathematics. This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein and Rami Shakarchi
The classical theory of functions of a real variable: point-set topology, limits and continuity, differentiation and Riemann integration, infinite series, uniform convergence