To provide an introduction to the basic notions and results of measure theory and how these are used in probability theory. In particular, we will gain understanding of sigma algebras, prove the monotone class theorem, introduce the concept of Lebesgue integrability, prove the monotone and dominated convergence theorem, Fatou's lemma, and Fubini's theorem, consider the concept of independent random variables, prove the Radon-Nikodym theorem, extensively discuss conditional expectations, discuss various types of convergence for random variables, prove discrete time martingale convergence theorems, prove the strong law of large numbers and the central limit theorem, discuss the relation between weak convergence and convergence of the characteristic function, and prove the existence of a Brownian motion.

The course is essentially self-contained, but the measure-theoretical basics (sigma-algebra, measurable space, Dynkin's lemma, Caratheodory's extension theorem, measurable functions, Lebesgue measure and -integral) are explained only very briefly. A student who is unfamiliar with this concepts will need to invest some extra time to succesfully complete the course.