Rick Durrett's book "Probability: Theory and Examples" is a very readable introduction to measure-theoretic probability, and has plenty of examples and exercises. This is the second text that I learned probability theory out of, and I thought it was quite good (I used Breiman first, and didn't enjoy it very much). As a bonus, there is a free .pdf version of the 4th edition (which will be published in a few months) available on his website for the time being.

A recent text covering similar material (which I admit I haven't read that fully) which looked good on a quick reading was "Probability Theory: A Comprehensive Course" by Klenke. It has a very nice selection of modern topics.