Most probability texts that do not use measure theory seemed to be geared toward engineers and the like, while more advanced texts already assume a strong background in measure theory and Lebesgue integration.
I'm not too familiar with measure theory, (my analysis background is limited to Rudin's PMA Ch. 1 - 10), so I'd appreciate some recommendations that do not assume an analysis background beyond Rudin's PMA. I'd be okay with a text that uses measure and Lebesgue theory as long as it sufficiently presents the material on its own.
Thank you very much.