Board photos & synopsis of material covered:

• lecture 1, Sept 25: Motivation for the setup of the Kolmogorov model of probability. (slides)
• lecture 2, Sept 28: Definitions of a σ-algebra and a probability measure. Properties of measures: monotonicity, subadditivity and continuity. Examples: discrete and continuum. The notion of a random variable. Equivalence of the problem of infinitely many (fair) coin tosses and a uniform random point in [0,1]. (photos)
• lecture 3, Sept 30: The measure extension problem. Definition of an outer measure and its construction from a covering class and a premeasure. The Carathéodory Extension Theorem and its proof. (photos)
• lecture 4, Oct 2: The Hahn-Kolmogorov theorem giving an extension of a premeasure on an algebra, subject to finite additivity and countable additivity assumptions, to a measure on a σ-algebra generated by the algebra. Further reduction to a premeasure on a semialgebra. The construction of Stieltjes measures on the line -- to be reviewed and continued next time. (photos)
• lecture 5, Oct 5: Validating the fact that the assignment (a,b] ↦ F(b)-F(a) gives a finitely additive and countably subaditive set function on the semialgebra of half-open intervals. The extension to a measure on the σ-algebra of Borel sets in ℝ. Onto the uniquness: The class of sets on which two finite measures coincide is a λ-system. Dynkin's π/λ-theorem and its proof. (photos)
• lecture 6, Oct 7: Examples of Stieltjes measures: Dirac measure, Lebesgue measure, counting measure on the integers, uniform measure on the ternary Cantor set. Every Carathéodory-measurable set is a countable intersection of countable unions of sets from covering class minus a null set. Lebesgue measurable sets. Checking measurability of functions by checking preimages of a generating class of sets. Existence of a continuous function that is not Lebesgue measurable. (photos)
• lecture 7, Oct 9: Sums and products, as well as measurable functions of real-valued random variables are random variables. Suprema, infima of sequences of random variable are random variables. The limit random variable. Simple random variables and their use in approximation; the Monotone Class Theorem. Prove that a σ(X)-measurable r.v. necessarily takes the form f(X), for a measurable function f (photos)
• lecture 8, Oct 12: Unsigned Lebesgue integral: motivation by Riemann's theory and Cavalieri principle. Integral of simple function and its independence of representation. Unsigned integral ∫ fdμ of non-negative f as the supremum over ∫ φdμ over simple φ such that 0≤φ≤f. Monotonicity, positive homogeneity and superadditivity of the unsigned integral. The need for measurability. Proof of additivity of bounded functions supported on finite-measure set. A key approximation lemma. (photos)
• lecture 9, Oct 14: Markov's inequality. Proof of additivity of unsigned Lebesgue integral. The Monotone Convergence Theorem. Commuting infinite sums with integrals of non-negative functions. Using integral to define new measures. Fatou's lemma. The signed Lebesgue integral, its additivity and homogeneity. The statement of the Dominated Convergence Theorem. (photos)
• lecture 10, Oct 16: Proof of the Dominated Convergence Theorem (DCT) via Fatou's lemma. Need a lemma that an absolutely integrable function is integrable. Bounded Convergence Theorem as a corollary of DCT. Further inequalities: Cauchy-Schwarz and Hölder's inequality. Back to probability: Expectation as the Lebesgue integral of the random variable. Change of variables formula for a measurable function of a random variable. Jensen's inequality.(photos)
• lecture 11, Oct 19: Independence. First, pairwise independence, then independence of arbitrary collections of events. Examples and counterexamples. Independence of random variables and σ-algebras. Independence of generating π-systems suffices. Relation of independence to product measures. The (finite-)product space with the product σ-algebra. Existence and uniqueness of a product measure for finite products. (slides)