PROGRAM:

1. σ-algebras, measures, measurable space. Theorem on the equivalent conditions for the
finitely additive set function to be a probability.
2. Borel σ-algebras, the measurable spaces (R, B(R)), (Rⁿ , B(Rⁿ )) and (R^∞ , B(R^∞ )).
3. Probability measures on measurable spaces. Theorem on one-to-one correspondence
between the probabilities and the distributions.
4. Measures: discrete, continuous, singular.
5. Random variables. Lemma: Borel function of a random variable is a random variable.
6. Theorem on the limits of the sequences of the extended random variables.
7. F_ξ  σ-algebra. Theorem on the representation of a F_ξ -measurable random variable.
8. Random elements. Definition of the independence. The necessary and sufficient
conditions of the independence.
9. Lebesgue integral: definition, properties.
10. Theorem on monotone convergence (without proof)
11. Fatou’s Lemma (without proof)
12. Lebesgue’s Theorem on Dominated convergence (without proof)
13. Chebyshev’s, Cauchy-Bunyakovskii and Jensen’s inequalities.
14. Lyapunov’s, Hölder’s and Minkowski’s inequalities.
15. Theorem on change of variables in a Lebesgue integral.
16. Fubini’s theorem (without proof)
17. Conditional expectation with respect to a σ-algebra: definition and properties.
18. Characteristic functions. Uniqueness theorem.
19. Helly’s Theorem.
20. Prokhorov’s Theorem.
21. Continuity Theorem.
22. Central Limit Theorem and Law of Large Number for i.i.d. random variables.
23. Zero-or-One Laws.