This course contains a part of the common culture of mathematics that every graduate student in mathematics should study and every mathematician should know. It represents the culminating point of view on the real number system that began back in calculus, and is required background for anyone wishing to understand current research in analysis, probability, stochastic processes, and differential equations.

The purpose of the course is to give the student a foundation for research in analysis or further study in applications.

In this first semester we will aim to discuss most of the core topics, including sigma-algebras and the monotone class theorem, Caratheodory's construction of Lebesgue measure on the real line, measurable functions and Littlewood's three principles, Lebesgue integration and convergence theorems, bounded variation and absolute continuity, n-dimensional Lebesgue measure and integration, probability measures, conditional expectation and independence, signed measures and the Radon-Nikodym theorem, product measures and Fubini's theorem, L^p spaces as time permits.

In the second semester we will finish discussion of the core topics and address some further topics from among Fourier Analysis, distribution theory, weak convergence and the central limit theorem, geometric measure theory, selected topics in probabiltiy and stochastic processes.