I'm looking for a book that serves as an introduction to probability theory. I barely know anything about this branch of mathematics but if possible I'd like it to involve plenty of analysis since that's what i've studied a bit of thus far. I'd like it to have a lot assignments as well. I'd prefer if it's more focused on pure mathematics and I want plenty of rigorous proofs.
I heard from my friend who is a PhD student doing research in Probability, that he recommends "Real Analysis and Probability", by Dudley. Here is its amazon page .
He said it is clear, detailed, thorough, and doesn't skip steps. Also, it helped him transition from "undergraduate" probability to measure-theoretic probability. From his comments and the amazon reviews, it seems to match your criteria of "plenty of analysis", "focused on pure mathematics", and "plenty of rigorous proofs". I haven't read it yet, but it's definitely on my reading list.