A First Look at Rigorous Probability Theory (2nd Edition)
by Jeffrey S. Rosenthal · World Scientific
Graduate textbook that constructs probability triples (Omega, F, P) from scratch, proving the extension theorem guaranteeing such spaces exist. About 275 exercises with full proofs. Readers finish able to state and verify the Kolmogorov axioms on uncountable sample spaces.
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275A, Notes 0: Foundations of Probability Theory
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.
Foundations of the Theory of Probability (Second English Edition)
The 1933 monograph that introduced the axioms of probability, in Nathan Morrison's English translation. Ninety-six pages defining elementary probability fields, extending them to infinite fields, and deriving random variables and expectation from set-theoretic first principles. or 'Kolmogorov axioms' the Source Truth is Kolmogorov, and it is unusually accessible: 96 pages, cheap, and readable by anyone who has seen basic set theory. Reading it after Rosenthal shows the learner how little machinery the axioms actually require.
Probability with Martingales
A 251-page Cambridge course that introduces measure theory then proves Kolmogorov's strong law and the three-series theorem by martingale arguments, reaching the central limit theorem via characteristic functions. Exercises carry much of the content.
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.