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], Chapter 1 |
| 25.2. | [IK] Proof of countable additivity for the Lebesgue measure. Terminology: null set, almost everywhere. Higher dimensional Lebesgue measure. General definition of measure and of measure space. Examples: counting measure, Dirac measure. Generalized definition of partition of a set. Simple function, integral of simple function. Integral of nonnegative function. A particular approximation of a nonnegative function by a simple function (as a motivation for why we only want to define integrals for measurable functions). Measurable functions, integral of a measurable function. Integrable functions. Lemma (without proof): for a bounded function f defined on a set of finite measure, supremum of the integrals of lower approximations by simple functions equals infimum of the integrals of upper approximations by simple functions, if and only if f is measurable. | [KMS], Chapter 1 |
| 4.3. | [IK] Measurable functions defined on general measure spaces (with codomain being the real line). In this context: simple functions, integral. Properties of integral. Complete measure spaces (and why we like them). Fatou's lemma. Monotone convergence theorem. Linearity of integral. Littlewood principles: mentioned informally. | [KMS], Chapter 1 |
| 11. 3. | [IK] Dominated convergence theorem. Product of measure spaces. Sigma-finite measure spaces. Fubini theorem, Tonelli theorem. Definition of probability space. Observation that the axioms are the same as for measure space, with the added requirement that probability of the whole set is 1. Motivation for the notion of density. In general, if we have a nonnegative measurable function f defined on a measure space, we may use it as "density" to define a new measure. This way, we get, e.g., the Gaussian measure from the Lebesgue measure. Motivation and definition of expected value. Normal distribution, computation of the constant of the density function. | [KMS], Chapter 2.3 |