Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion.
Good solid book on the topic. Plenty of helpful exercises. I used this for a class along with Probability: Theory and Examples and they complement each other well. We used this one more at the start, since it covers the underlying measure theory in more detail. The font for some of the symbols made them a bit hard to read.