- Intro; Branching process example.
- What is a probability space? Algebras, sigma-algebras and measures.
- Dynkin's lemma; uniqueness of extension.
[Supplementary reading: A non-measurable set from coin flips (optional).]
- Caratheodory's extension theorem.
- Lebesgue measure; completion of a measure space.
- Events; concrete probability spaces; Fatou for sets; Borel--Cantelli Lemma.
- Measurable functions; random varibales; generated sigma-algebras.
- Distribution functions; Lebesgue-Stieltjes measure; "Granularity".
- Indepenence: events, random variables, sigma-algebras; sufficient conditions for independence; Second Borel--Cantelli lemma.
- Sequences of independent random variables; Markov chains by hand; an aside on stochastic processes and continuity.
[Supplementary reading: The Kolmogorov extension theorem (optional)]
- Integration: definitions, basic properties; monotone convergence theorem.
- Fatou's lemma; Dominated convergence; Riemann versus Lebesgue integration.
- Integrals over subsets, change of measure, and a first glimpse of Radon--Nikodym. PDFs and the "elementary formula" for expectation.
- Expectation: Markov, Jensen, Hölder, Minkowski, and L^p-spaces.
- Cauchy--Schwartz and the metric structure of L^2.
- Laws of large numbers 1.
- Laws of large numbers 2.
[Supplementary notes to be posted.]
- Product measure; the monotone class theorem.
- Fubini's theorem.
- Independence and product measure; product spaces and separability; cylinder sets and an aside on continuity.
Cylinder sets and continuity (required, except for the section on Kolmogorov extension theorem).]
- Conditional expectation: the defining property, existence, almost sure uniqueness.
- Fundamental properties of conditional expectation.
- Conditional PDFs, conditional probabilities, conditional distributions.
Conditional distributions (required).]
- A second glimpse of Radon--Nikodym, and a first glimpse of Martingales.