Books on measure theory

Question

Can someone kindly suggested a good book on measure theory? Taking into consideration a good treatment of the abstract measures and Caratheodory approach.

Answer 1

I've used Elstrodt's "Maß- und Integrationstheorie" a lot, and believe it to be an excellent introduction to the topic. The book is german, though - I don't know if there's an english translation.

Answer 2

Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts) [Hardcover] Gerald B. Folland (Author)

http://www.amazon.com/Real-Analysis-Techniques-Applications-Mathematics/dp/0471317160

Answer 3

I really found Tao's book An Introduction to Measure Theory really quite excellent. It provides a lot of motivation as well as a lot of foundation to the theory of measure. Most books just start talking about $\sigma$-algebras without really explaining the motivation, etc. With Tao's book he works up the development of the Lebesgue integral on $\mathbb{R}^n$ and then gradually moves to the parallel development of Lebesgue integral on abstract measure spaces, using $\sigma$-algebra. This is a really effective treatment because the reader understands the relationship between borel sets on $\mathbb{R}^n$ and the sets in a $\sigma$-algebra.