This is a fantastic book on measure theory. The focus is on measure theory on its own right and not on probability. I was lucky to come across this book while canvassing the measure theory books at our library. I looked at the books by Billingsley, Halmos, Chung, Resnick, Rao, Rudin, Pollard, Dudley, Nielson, Stroock, Williams, Pitt, and many others. Hand-down, Vestrup is the best.
I believe after scrutinizing so many books, I have a very good baseline to judge Vestrup's work. Here are a few specific reasons:
(1) If you don't like detail and revel in banging your head against the walls to figure out the skipped details in Billingsley, this is not the book for you. But If you are a first timer to measure theory, this is as good as it will get; All the major results of measure theory are presented in detailed and clear manner with few skipped details and few not-so-obvious "it is obvious" remarks.
(2) Vestrup has a lot of exercises with lots of helpful hints. Some problems at first appear to be long and intimidating till you look closely and discover that Vestrup leads you through the problems with his hints.
(3) Certain topics central to understanding of measure theory were given cursory coverage by most of the books mentioned above. Not Vestrup. For example, Vestrup devotes a whole chapter to extensions. This is just one example of many central ideas Vestrup develops meticulously and painstakingly.
This book is fairly new and I think its popularity will grow as more students and professionals discover it. I suppose the only criticism I have is that the typesetting can be improved (second edition maybe?)
There are a few other good books (Ash, Bartle, and Royden) that are out there that you may consider but again Vestrup trumps them all. Whatever you decide on, I strongly warn against using Billingsley.