5. Some topics from measure theory.
Exterior measures. Construction of a measure. Caratheodory theorem. Lebesgue measure. Main properties of positive measures. Measurable functions/randon variables. Integrable functions. Monotone convergence theorem, Fatou's lemma, dominated convergence theorem. L^p spaces. L^2 viewed as an Hilbert space.
6. Independent random variables (r.v.).
7. Probability distribution on R.
8. Probability distributions on R^n.
9. Characteristic functions and their properties.
10. Sums of independent random variables.
11. Gaussian r.v.
12. Convergence of r.v. (convergence in probability, weak convergence)