Semi-algebras and sigma-algebras of events, Kolmogorov?s axioms of probability, consequences thereof, probability spaces, measurability, random variables as measurable mappings, random vectors, probability measures induced on Borel sigma-algebras by random vectors, distributions and distribution functions, extension of probability measure starting by semi-algebras, mathematical expectation, expected values of non-negative simple, non-negative and general random variables, properties, conditional distributions and independence, Borel-Cantelli lemma, conditional expectation given a sub sigma-algebra, Radon-Nikodym theorem, different modes of convergence, almost sure convergence, convergence in probability, convergence in L^p, convergence in distribution, different implications between them, characteristic functions, inversion formulas, relation to convergence concepts, the weak and the strong law of large numbers, central limit theorem.

1. Patrick Billingsley, Probability and Measure,
Volume 245 of the Wiley Series in Probability and Statistics, 3rd ed.
2. Erhan Çınlar, Probability and Stochastics, Volume 261 of Graduate Texts in Mathematics, Springer.

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