275A, Notes 0: Foundations of Probability Theory
by Terence Tao · UCLA (Math 275A course notes)
Every textbook defines a probability space; almost none explain why that definition and not another, or what it means that the sample space is largely arbitrary and can be extended at will. This is the best free treatment of exactly that question, from an author with unimpeachable authority, and it is the piece that turns the definition from a formality into a mental model. Opening lecture notes from Tao's UCLA graduate probability course, arguing why the (Omega, F, P) formalism is set up the way it is: extending sample spaces, which statements survive a change of underlying space, and what 'probabilistic thinking' formally means.
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275A: Probability Theory - Terence Tao's course notes
Where Durrett gives one canonical proof, Tao gives three - characteristic functions, the moment method, and the Lindeberg exchange - and explains what each buys you. That comparative view is what turns the CLT from a memorised theorem into a technique you can transfer. Six sets of graduate lecture notes with exercises, building from measure-theoretic foundations to Notes 3 on the weak and strong laws, Notes 4 on the central limit theorem, and Notes 5 on variants including Berry-Esseen and stable laws.