This will be a course in measure theory. We develop simultaneously the theory of Lebesgue measure/integration on the reals and the measure theory needed to do probability. Standard results about convergence and inequalities involving integrals will be done. We will use Real Analysis by H.L. Royden as a text, as well as supplementary notes on probability by Greg Lawler. There will also be an introduction to Fourier analysis including a proof of the central limit theorem using Fourier analysis.

Prerequisite: Mathematics 204 or the equivalent; this includes a rigorous introduction to to the real number system.