Book on "Measure and integration" for starters. This semester I have a course on Measure and Integration. I'd like you to recommend me some books.
 A: I cannot recommend highly enough the book Lebesgue Integration on Euclidean Space by Frank Jones.  
A: How about:
Measure Theory by Donald L. Cohn
A: I think these two would be exactly what you are looking for. They helped me a great deal starting out.
(1) Measure Theory: A First Course.
(2) Lebesgue Integration On Euclidean Space (as mentioned by Tim kinsella)
These two books seemed the "easiest" and clearest to me.  Perfect as starters.
A: I think that Royden's book on real analysis is also a good starting point. It deals with measure theory on two levels: (1) Lebesgue measure on $\mathbb{R}$, and (2) abstract measure theory.
Another choice could be Halmos' Measure theory, but it is a bit old-fashioned and probably less complete.
A: Terence Tao's An Introduction to Measure Theory. I like his exposition very much as he first constructs the Jordan measure (which is very intuitive if one is used to idea that one can calculate the area of a set by "approximating" it with sets of a more elementary nature like squares or polygons). He then connects the Jordan measure to the Riemann Integral and explains why there is a need to develop a more general (lebesgue) measure/Integral. He then moves on to integration theory and defines the lebesgue integral. Once you have struggled through the first sections of the book and gotten used to the ideas presented there, I think the leap to abstract measure space theory (which is presented in one of the sections) is short. The rest of the book is devoted to convergence/differentiation theorems and construction of measures.  What I liked with the book is that he does not start off with abstract measure theory, constructions of measures (like Caratheodorys theorem) and all that  without little or no motivation. Another nice feature is that he leaves many of the easy propositions as exercises. Also, last but not least, there is a very nice section on problem solving techniques, which could be very useful.
A: The following two books are recommended:


*

*Walter Rudin, Real and Complex Analysis.

*G. B. Folland, Real Analysis.
