This year we will focus on measure theory -- as a background for probability. Also on geometry in high dimensions and a bit of functional analysis.
| Date | Topic | Literature |
|---|---|---|
| 18. 2. | [IK] Motivation for why we need the notion of Lebesgue measure (i.e., a generalization of the length of an interval to more general sets): integrating more exotic functions, probability. Four natural properties that we would like this measure to satisfy (it is defined on all subsets of reals, for intervals it agrees with their length, it is translation invariant and countably additive) . Definition of outer Lebesgue measure and proof that it has three-and-half of the four properties: instead of countable additivity, we could prove only countable subadditivity. Sadly, there is a counterexample to countable additivity: the Vitali set. And tragically, it can be proved our whole task is hopeless: that no set function on \(\mathbb{R}\) satisfies all 4 properties. We give up on the first property, define (Lebesgue) measurable sets and restrict the outer measure to this family. Definition of \(\sigma\)-algebra and mentioned that measurable sets form a \(\sigma\)-algebra (proof left as an exercise). Definition of smallest \(\sigma\)-algebra generated by a set. Borel sets. Mentioned that Borel sets are measurable (proof left as an exercise). | [KMS], 1.1 |