Most posts here asked for a probability textbook which does not assume measure-theoretic background.
However, I have a quite concrete background of measure-theory and am looking for a probability textbook which is very measure-theoretic and written for pure mathematicians, not engineers.
(I am a third-year graduate student and I have studied at least 4 different real-analysis textbooks including Folland’s, Stein’s, Royden’s, Rudin’s and etc. However, I have never studied probability theory before, and I need to study this theory now.)
Many years ago, when I did my preparation for the measure theory exam, one of the main sources for my studies was Measure and Integration Theory by Heinz Bauer. It was perfect for starters providing an easy to read, solid and instructive introduction into the subject.
In fact the book about measure theory was an integral part of the Probability Theory when the first editions have been published. But it has been extracted as many students wanted to have this content in a book by itself. When H. Bauer followed these suggestions, he also did a rework of the Probability Theory but with strong focus on his Measure Theory.