Real Analysis II

The class is aimed to give a rigorous exposition of the measure theory and Lebesgue integration. These techniques are widely used in modern analysis, in particular, in mathematical probability. Tentative topics are: Families of subsets (Sigma-algebras, Rings, Semi-rings and Dynkin systems); Measure; Caratheodorys extension theorem; Measurable functions; Various types of convergence (almost everywhere, almost uniformly, in measure); Lebesgue integral; Fatou, monotone convergence and dominated convergence theorems; Product measures, Fubinis theorem; Signed measures; Absolute continuity and Radon-Nikodim theorem. Prerequisites: 92.501 Real Analysis or equivalent