Reddit reviews Probability and Measure

"Probability theory is just a branch of measure theory" is something analysts say to annoy probabilists. It has just enough truth to be annoying but not enough to truly sting. After all the first third of the typical graduate level probability text, such as Chung or Billingsley is mostly measure theory. But then probability theory goes off in its own direction with some of the deepest theorems in mathematics, which have no analogs in any other part of mathematics.

For undergrad probability, Pitman's book or Ross's two books here and here.

For graduate probability, Billingsley (h/t /u/DCI_John_Luther), Williams or Durrett.

1. Ross - First Course in Probability (Calculus based probability, undergraduate level, good introduction to probability)
   
   http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
   
   
2. Rice - Mathematical Statistics (introduction to statistics, focuses on applications with data, great book, includes good probability review)
   
   http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
   
   
3. Billingsley - Probability and Measure (graduate, measure-theoretic probability)
   
   http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
   
   
4. Bickel & Doksum - Mathematical Statistics (graduate level statistical theory, much more theoretical than Rice, can be a difficult book to learn from but it is a great reference)
   
   http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
   
   EDIT:
   
   Most likely Rice will be the best book for a comprehensive look at prob/stat, and it is sufficiently technical.
   

Awesome! As mentioned, Rudin, Folland, and Royden are the gold standards of measure theory, at least from what I have heard from professors and the internet. I'm sure other people have found other good ones! Another few I somewhat enjoy are Capinski and Kopp and Dudley, as those are more based on developing probability theory. Two of my professors also suggested Billingsley, though I have not really had a good chance to look at it yet. They suggested that one to me after I specifically told them I want to learn measure theory for its own right as well as onto developing probability theory. What is your background in terms of analysis/topology? Also, I am teaching myself basic measure theory (measures, integration, L^p spaces), then I think that should be enough to look into advanced probability. Feel free to PM me if you need some help finding some of these books! I prefer approaching this from the pure math side, so mathematical statistics gets a bit too dense for me, but either way, I would look at probability then try to apply it to statistics, especially at a graduate level. But who am I to be doling out advice?!

*Edit: supplied a bit more context.

I have always been fond of Probability and Measure by Patrick Billingsley.