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What book would you recommend me to read about measure theory and especially the following:

Measure and outer meansure, Borel sets, the outer Lebesgue measure. The Cantor set. Properties of Lebesgue measure (translation invariance, completeness, regularity, uniqueness). Steinhaus theorem, non-Lebesgue measurable sets. Measurable functions, integrable functions, convergence theorems. Elementary theory of Hilbert spaces. Complex measures, the Radon-Nikodym theorem. The maximal function Hardy-Littlewood. Differentiation of measures and functions. Product of measures. The Fubini theorem. Change of variable. Polar coordinates. Convolutions.

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    $\begingroup$ Folland's book Real Analysis is a standard choice. I feel that it is very technical and probably not best for a first pass depending on your "mathematical maturity", but it has all of those things. I found it difficult to go through the first time, but appreciated it going through it again and like it a lot now. $\endgroup$ – dannum Jan 18 '15 at 2:23
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    $\begingroup$ I think most of this is covered by Bartle's book. It is a compact Wiley Classics book. $\endgroup$ – Jas Ter Jan 18 '15 at 21:47
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    $\begingroup$ What you're writing here look like it was taken from a syllabus or some kind of exam requirements. I think you should say which one, and what are you trying to accomplish. If it is the former, you should also ask the lecturer for some recommendations. $\endgroup$ – tomasz Jan 18 '15 at 22:01
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I would recommend you to start with Terence Tao's An Introduction to Measure Theory.It will cover the basic areas of measure theory .Moreover it is very well written.Not sure that it will cover every topic of yours but will cover most of them.Also the exercises are mind blowing.Try it, you will surely feel the difference from other books

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  • $\begingroup$ Tao's book is hard to read as a novice to measure theory I believe $\endgroup$ – Permian Jan 1 '16 at 15:31
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Gerald Folland's Real Analysis book is a superb choice for learning measure theory and elementary functional analysis. It covers all the topics you listed.

It is a bit more difficult and abstract than most other introductory textbook (such as Royden), but it is well worth it (and yes, it is actually suitable for a first-time learner, because I was). If you want a solid foundation on measure theory, this is the single book you should own.

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I would start with Real Analysis by Royden. I find it very well-written and the problems are really good. It covers most of the areas you mention.

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    $\begingroup$ +1. With the goals in the OP, I would recommend trying to study from the second half about general measure theory and jumping back to the real line sections when you have trouble understanding. If you find yourself jumping back repeatedly, just study straight through. $\endgroup$ – Ian Jan 18 '15 at 21:47
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A slightly different approach would be Bogachev's two-volume 'Measure Theory', especially if you want to focus more on measures rather than general analysis. It covers all the topics you listed in depth except Hilbert spaces. It does, however, have a chapter on $L^p$-spaces.

The good thing about this book is that it has plenty of exercises and supplemental sections (taking nearly half the book) that cover aspects of measure theory not usually covered in a book on analysis, and is really meant for someone who wants to become intimate with measure theory. At the very least it is a great reference to delve into more depth.

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