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This book has 1 recommendation
Nassim Nicholas Taleb (Flaneur)
I know which books I value when I end up buying a second copy after losing the first one. This book gives a complete overview of the basis of probability theory with some grounding in measure theory, and presents the main proofs. It is remarkable because of its concision and completeness: visibly prof Varadhan lectured from these notes and kept improving on them until we got this gem. There is not a single sentence too many, yet nothing is missing.
For those who don't know who he is, Varadhan stands as one of the greatest probabilists of all time. Learning probability from him is like learning from Aristotle. Varadhan has two other similar volumes one covering stochastic processes the other into the theory of large deviations (though older than this current text).
The book on Stochastic Processes should be paired with this one.
This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences, USA. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation.