Prerequisites

None (but it does help to have some basic understanding of (functional) analysis; e.g. convergence, (uniform) continuity, metric spaces, Banach/Hilbert spaces). Also, although measure- and Lebesgue integration theory is built up "from scratch", the course is quite tough for those who have never seen measure theory before. In that case it may help to consult René Schilling's Measures, Integrals and Martingales, or Jeffrey Rosenthal's A First Look at Rigorous Probability Theory for some extra material on this topic.

Aim of the course

  • Discuss the measure- and Lebesgue integration theory that is relevant in probability theory.
  • Introduce some vital concepts in probability theory, such as conditional expectations, the Radon-Nikodym theorem, martingale convergence theorems, characteristic functions and why they are characteristic, the Brownian motion.
  • Provide rigorous proofs for two central convergence theorems in probability: the Strong Law of Large Numbers and the Central Limit Theorem.