Probability & Measure Theory

Rating:
Summary: Exceptionally Clear
Review: I first used this text in the earlier version, which comprises the first half of the book, in a one-year course in Hilbert Spaces and Lebesgue Measure theory when in the first year of grad school. The material is presented in a clearly written manner and the exposition is some of the clearest mathematical writing I've seen in a subject which is replete with textbooks.

Anyone who wants to be inaugurated into the "mysteries" of measure theory and the fine points of the rigorous theory of stochastic processes and the Ito integral, will do himself or herself a favor by using this text. If it is not assigned to your class and you have the extra cash, order it anyway. It is also well-suited for self-study.

Rating:
Summary: The best introduction to probability and measure
Review: The book very nicely develops the basics of measure theory from a probability perspective (e.g. includes Caratheodory extension theorem, Lebesgue-Stieltjes measures, weak convergence and Kolmogorov extension theorem). It then gives a brief introduction to functional analysis and proceeds to probability theory, martingales and concludes with brownian motion and stochastic integration.

All standard results are given and the book is self-contained. It is a concise, yet readable introduction to this area (less concise then Rudin, Williams but more than Billingsly). An excellent feature of this book is that full solutions to some of the exercises are provided at the end. This makes this book ideal for self-study. The only prerequisite for this book is elementary real analysis (say chapters 1-7 of Rudin's principles of mathematical analysis).

There are other excellent books on measure theory (Rudin, Royden), but if you are interested in measure theory from a probabilistic view this is the book to choose.
As far as a probability textbook, it is clearer and more readable than Billingsly, Chung, Williams and Durrett.