Reddit mentions of Real Analysis

>Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Rudin, Principles of Mathematical Analysis by Kolmogorov & Fomin

Permuted the authors :P. That being said, Rudin's first book I don't think is the best introduction to real analysis (and Kolmogorov's I think is better as a bridge towards functional analysis). My recommendation for a first real analysis book is Real Analysis by N.L. Carothers.

If you would like to become an expert in probability theory, you need to have a solid ground in measure theory. I would suggest to study analysis out of Carothers. This covers most of what Rudin covers but I find it easier to read, and it goes into more detail about measure and Lebesgue integration on the real line. If you work through this, you'll have a solid background for heavier measure theory books and for upper level probability theory.

I used Carothers in my Real Analysis class, but if you're looking for a readable, introductory book which doesn't skip any steps, Bruckner is a great, free choice. I used this textbook in an Advanced Calculus class and really enjoyed reading it.

For a book that's less likely to defeat the reader (because even Baby Rudin is very tough), I'd like to recommend N. (Neal) L. Carothers - Real Analysis.

Metric spaces and Lebesgue measure on the real line only, but quite well written. Honestly, most of what's in there (on metric spaces) should probably be covered in your next analysis course, but owning more good books never hurt, right?