Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.
Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory. --back cover
Overall I'm quite happy with it, although then again I am comparing against Kallenberg so maybe my baseline is just skewed.
I also only read chapters 5 and 6 (convergence and conditionals), although I'd be surprised if earlier chapters were worse given the relatively easier material.
Almost any book on formal probability ends up referring to this book at some point. Therefore, it’s better to just begin with this one. It is worth to mention that it’s contained
This book succeeds at its apparent goal: be a single, self-contained work on measure-theoretic probability. Having worked through Billingsley's book, if someone asked me how to learn measure-theoretic probability my answer would be to work through a book on measure theory (say, Royden's Real Analysis) without probability, and then read Kolmogorov's seminal work Foundations of the Theory of Probability which applies measure theory to probability.