A Good Book for Mathematical Probability Theory

Question

I am from mathematical background, and I always hated the way they teach elementary probability theory in schools without giving any clue about measure theory.

I want a theoretical book in probability theory, which starts with measure theory and defines everything in mathematical terms. I am also not looking for a measure theory book. If there is such a book that instead of giving only an intuitive introduction to probability theory, it rather defines everything in details, I would be really grateful to know it.

I would be happy to hear about any reading experience. Being exact in mathematical language and maybe exercises.

Thanks a lot.

Answer 1

Read Probability Theory and Stochastic Processes with Applications by Oliver Knill.

Answer 2

You should definitely read Probability and Measure by Patrick Billingsley. The book introduces the probability theory simultaneously with the measure theory -- there is not any division like this: first measure theory, then probability theory. The language of the book is clear and understandable, but still rigorous. The book is rich in examples -- after every theorem, definition etc., there are usually two or three of them. Each section ends with a bunch of exercises to which there are hints and notes at the end of the book.

Answer 3

Try Probability with Martingales, which I found to be very helpful for self-study. It's a short read but nevertheless concise and uses measure theory to introduce the basic concepts of probability theory.