I think there is a gap in the suggested books... there are no real analysis books... it is true that once you learn analysis from Rudin's principles yuo can master a measure-theoretic book in probability that provides a self-contained introduction to real analysis... but i think it is worth to spend some months on the subject of real analysis per se.... also b/c it is a beautiful topic...

what do you think about, e.g.:

Royden - real analysis

Wheeden & Zygmund - Measure and integral - An introduction to real analysis

Folland - Real Analysis Modern techniques and their applications

Rudin - real & complex analysis