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

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    $\begingroup$ See here, here, or here. $\endgroup$ – Michael Greinecker Mar 5 '14 at 15:51
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    $\begingroup$ Here are a couple of books I don't think have been mentioned: Measure Theory and Integration by Michael E. Taylor (2006) is very readable and it touches on a lot of supplementary topics of importance in contemporary research. Measure Theory and Integration by M. M. Rao (2nd edition, 2004) is one place to go if you want to jump in the deep end of the pool right away. $\endgroup$ – Dave L. Renfro Mar 5 '14 at 20:29
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    $\begingroup$ @Dave thanks a lot i just checked the book by Micheal Taylor.. Seems very helps.. Thanks $\endgroup$ – Ibrahim Sarumi Mar 6 '14 at 15:50
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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

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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.

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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.

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