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These lecture notes are intended for a first semester of graduate-level measure-theoretic probability. Topics covered include: foundational material, independence, zero-one laws, laws of large numbers, weak convergence and the central limit theorem, conditional expectations and their properties, and martingales. The notes were used for a one-semester course at UW-Madison (Fall 2013) and two quarter courses at UCLA (Winter 2012, Winter 2011, Fall 2010, Winter 2010). The material is based largely on the following references:

- [D] Probability: Theory and Examples (4th Edition) by Durrett
- [W] Probability with Martingales by Williams
- [Sh] Probability by Shiryaev

The notes are divided into one-week worth of material (two 75-minute lectures):

- Notes 1: Measure-theoretic foundations I
- Notes 2: Measure-theoretic foundations II
- Notes 3: Modes of convergence
- Notes 4: Laws of large numbers
- Notes 5: More on the a.s. convergence of sums
- Notes 6: Weak convergence and characteristic functions
- Notes 7: CLT and Poisson Convergence
- Notes 8: Method of moments
- Notes 9: Infinitely divisible and stable laws
- Notes 10: Random walks
- Notes 11: Conditioning
- Notes 12: Martingales
- Notes 13: Branching processes
- Notes 14: Martingales in Lp
- Notes 15: UI Martingales

Last updated: December 2013.