If you want to go deeper into probability theory: I would also recommend Heinz Bauer's "Measure and integration theory". But I found it a bit dry (in German). Also Kai Lai Chung's "A Course in Probability Theory" is excellent.
If you're more interested in (functional) analysis and want just a short intro to measue theory: As an alternative to Rudin's "Real and Complex Analysis" I warmly recommend the recent book by Jürgen Jost "Postmodern Analysis", which includes an intro to PDE. I wish I had that when I was young... Right next to these in my library is Segal & Kunze "Integrals and Operators" and Robert Geroch's "Mathematical Physics" (no physics inside).