I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in Mathematics - as possible. What I mean by that is that the book should contain and rigorously address the following topics:
- Measure Theory (As a mathematical foundation for probability)
- It is of course fine if this theory is addressed with an emphasis on probability and not only for the sake of mathematical measure theory, although the latter would be great too.
- Introduction to Probability, i.e. the most common theory a student is exposed to when taking a first course in theoretic Probability. For example: distributions, expected value, modes of convergence, Borel Cantelli Lemmas, LLN, CLT, Gaussian Random Vectors
- More advanced topics such as: Conditional Expectation (defined through sigma-Algebras), Martingales, Markov Processes, Brownian Motion
I want it to be one book so I can carry a physical copy of it with me and work through the material in my spare time.
- The book by Jean-Francois Le Gall which can be found here: https://www.math.u-psud.fr/~jflegall/IPPA2.pdf but (unfortunately for me) is written in French.
- Rick Durrett's book on Probability which can be found here https://services.math.duke.edu/~rtd/PTE/pte.html - the critique available for this book seems a bit mixed, uncertain about how to weigh that.
I am well aware that it's not easy to meet all of the above criteria simultaneously, but I would be grateful for any recommendation.