- Know and determine properties of systems of sets such as semi rings, sigma-algebra and Dynkin systems;
- Know the definition of general measures and measurable maps and understand their properties;
- Know the definition of the Lebesgue measure and its special properties;
- Comprehend classical theorems such as the theorem on uniqueness of measures and Carathéodory's extension theorem;
- Understand the construction of integrals w.r.t. general measures and know the standard procedure;
- Know and apply important convergence theorems, in particular the theorem by Beppo-Levi, the lemma of Fatou, the theorems of monotone and dominated convergence;
- Understand the definition of L_p spaces and their properties and prove elementary facts about them independently;
- Comprehend the construction of product measures;
- Apply Fubini's theorem to analyse and compute multi-dimensional integrals;
- Understand the connection between integrals and measures with densities as well as the theorem of Radon-Nikodým.
|
Systems of sets, maps on sets, measures, theorem on uniqueness of measures, Carathéodory's extension theorem, Lebesgues measure in R^n, measurable maps, integration w.r.t. general measures, convergence theorems, L_p spaces, product measures, Fubini's theorem, measures with densities, theorem of Radon-Nikodým
|
