Here are some typeset notes from probability classes I took at Columbia.

Classical Probability

[Notes] Measure Theory (construction of Lebesgue measure, measurable functions, sigma algebras, convergence). Notions of convergence (almost everywhere, in probability, LP, vague, etc). Interchange of expectation and limit (Monotone Convergence, Fatou’s Lemma, Dominated Convergence). Borel-Cantelli and other Zero-One Laws. Weak vs Strong of Large Numers, Central Limit Theorem. Markov Chain Convergence. Inequalities for Lp spaces.

Advanced Probability

[Notes] Martingales (uniformly integrable, square-integrable), stopping times, Markov Chains and harmonicity, optimal stopping, martingale inequalities, stochastic approximation, construction and path properties of Brownian motion, Ito integration, diffusion processes, stochastic control, portfolio theory, representation theorems (Girsanov, Dambis-Dubins-Schwarz).