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Lecturers: Michael Multerer, Jacopo Quizi
Course overview
This course develops probability from the language of measure theory. The summary starts from the set-theoretic and analytic tools needed for the subject, then builds probability spaces, measurable maps, Lebesgue measure, integration, independence, and the main convergence results used in probability.
Main topics
- useful notions: conditional probability, rational numbers, telescopic identities, open rectangles, set operations, and convergence of functions
- probability spaces: measurable spaces, sigma-algebras, measures, and probability spaces
- Lebesgue measure: Borel sets, construction of the Lebesgue-Borel measure, and null sets
- random variables: measurable functions and constructions of measurable functions
- independence: independent random variables and the Borel-Cantelli lemmas
- the integral: construction, properties, Lp-spaces, and convergence theorems
- Laws of Large Numbers: weak and strong laws
- Central Limit Theorem