I am looking for a book that deals with the fundamentals of probability (e.g. probability spaces, distribution functions, expectation, etc.) but from a measure-theoretical perspective (e.g. defining the expectation in terms of the integral w.r.t. a measure derived from the CDF).
Probability: I have already taken basic and not-so-basic probability classes. These have dealt with the fundamentals of probability in a rather rigorous manner, so I am quite familiar with it. However, these courses have never involved measure theory more than simply discussing the Borel $\sigma$-algebra.
Measure theory: I have already taken a course that has dealt with the lebesgue measure very rigorously. I have also done some reading on my own about some basic measure theory (in particular the first chapter of Bogachev's Measure Theory).
What I'm looking for: I always found it bizarre how expectation was defined differently for discrete and continuous random variables (and RVs that were neither were totally ignored). I then learned that this could be resolved by defining measures with the distribution functions, and integrating with respect to these measures.
However, I never saw this explained fully, and I have not been able to find a book on probability that explains this either. I am looking for a book that starts from the very basics of probability and measure theory, and builds up probability using these tools. Preferably the book will also show how these general definitions become the simple ones we all know in special cases (e.g. if the measure is discrete, the expecation is just a sum).
Thank you for any recommendations!