This course will follow the textbook:
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Durrett, Rick. Probability: Theory and Examples. 4th ed. Cambridge University Press, 2010. ISBN: 9780521765398. [Preview with Google Books]
I use the same notation as Durrett whenever possible.
In addition to the main textbook, there are many excellent textbooks and sets of lecture notes that cover the material of this course, several written by people right here at MIT. The course material is contained in the union of the following online texts for first-year graduate probability courses:
- S.R.S. Varadhan's lecture notes
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Amir Dembo's lecture notes (PDF)
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Rick Durrett's book at CiteSeer (PDF) or at Amazon and here is a recently updated version (PDF) from Durrett's web page.
- Noel Vaillant's www.probability.net tutorials
- Dmitry Panchenko's notes for an earlier rendition of 18.175.
A gentler introduction to some of the material in the course appears in David Gamarnik's notes. For a more general analysis reference, there is also the online text Applied Analysis by Hunter and Nachtergaele.
Other excellent graduate probability books (that I don't think have been posted online, at least not by the authors) include (but are obviously not limited to):
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Billingsley, Patrick. Probability and Measure. Anniv. ed. Wiley, 2012. ISBN: 9781118122372. [Preview with Google Books]
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Dudley, R. M. Real Analysis and Probability. Cambridge University Press, 2002. ISBN: 9780521007542. [Preview with Google Books]
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Stroock, Daniel. Probability Theory: An Analytic View. 2nd ed. Cambridge University Press, 2010. ISBN: 9780431087023. [Preview with Google Books]
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Williams, David. Probability with Martingales. China Press, 2008. ISBN: 9787506292511.
There's a lot of overlap between these books, but you'll develop strong opinions if you spend much time with them. Here is one person's rated list of graduate probability books. (You probably won't agree with the list author's opinions, but it's still a nice list.)