Summary of lectures
- Lecture 1, 20/1: Section 2 in Schilling. We covered some operations of sets, and defined how one can compare the cardinalities of different sets using injections, surjections, and bijections. The main takeaway was that the cardinality of the rational number Q is countable, and so too is Q^d for any natural number d.
- Lecture 2, 21/1: Section 3 in Schilling. We motivated what a measure is (more on that next week), and defined sigma-algebras, the power set of X. For a family of sets G \subset X, we defined sigma(G), the sigma-algebra generated by G. We looked at the Borel sigma-algebra on R^n, and showed that the sigma-algebra generated by the family of half-open rectangles is equal to the Borel sigma-algebra.
- Lecture 3, 27/1: Section 4 in Schilling. We covered the properties of measures in Proposition 4.3, gave some examples of measures and presented Lemma 4.8: for additive functions with mu(emptySet) = 0, mu is a measure iff it is continuous from below.
- Lecture 4, 28/1: Section 5 in Schilling. We looked at Dynkin systems proved connections between Dynkin systems and sigma algebras generated by a family of subsets calligraphic \subset Powerset(X). And we proved Theorem 5.7 on uniqueness of measures.
Published Jan. 20, 2025 5:26 PM
- Last modified Jan. 28, 2025 2:00 PM