Good textbooks for self study in measure theory

[danny1997]

Hey all! I'm currently working on my masters in statistics, and I am hoping to next fall apply to PhD programs in statistics. At the moment I have not taken a course in measure theory, so I am hoping to study it myself over the next few months. Does anyone know of any good textbooks for this? I would be even more interested in one that directly relates to probability if a good one exists.

[Jdriii]

"Measure, Integral and Probability" by Kopp and Capinski is a good choice. It was written with self study in mind, and have a probabilistic bent to it's teaching.

[bayessays]

Just so you know, the overwhelming majority of statistics PhDs will never take a course devoted to measure theory in their life.  Most PhD programs use a book like Resnick's Probability Path to teach probability with measure theory, which doesn't require having taken a measure theory course.

[jelquiades]

I second Kopp and Capinski. Not only is it exactly the material you're looking for, the book contains detailed solutions to every problem, crucial for self study. Plus it has bonus sections on applications to finance, if you're into that.

[BL250604]

Probability and Measure, Billingsley is what my PhD program uses for our measure theory course.

[J456]

I haven't read it but it may be worthwhile to look into A Basic Course in Measure and Probability: Theory for Applications. I read somewhere that this book evolved from lectures//courses at UNC Chapel Hill. Over there, they seem to spend one whole semester on measure theory with an eye towards probability. So unlike  Durrett's book, you aren't expected to know measure theory beforehand, and unlike other books, measure theory isn't introduced hastily to just get you started. May be an ideal compromise, and the book seems readable, unlike the other standard but but difficult books like Durett (too terse) and Billingsley  (encyclopedic). 

[captivatingCA]

I think the recommendations for textbooks have been good so far, but in case you want to supplement your studying with lectures, Bilkent University has online lectures for probability theory available on YouTube. The course is tailored for industrial engineering PhD students, so it covers all the prior knowledge you need. There's a lot of content (nearly 50 hour-long lectures), and the lecturer is great. 

https://www.youtube.com/playlist?list=PL5B3KLQNAC5jT6yjV1199ji1zUy1YUp6P