Probability Theory

σ-fields and Probability Spaces: Probability as a measure. Existence and uniqueness (Carathéodory theorem) of probability measures. Independence: Sequences of events: 1st and 2nd Borel-Cantelli lemmas. Random Variables: Measurable functions. Expectation of a random variable as Lebesgue integral. Characteristic function of a random variable. Modes of Convergence: P. Lévy theorem. Sums of Independent Random Variables: Strong Law of Large Numbers, Central Limit Theorem. Conditional expectation of a random variable. Discrete Time Martingales.