Integration and measure theories, a reference list

Question

This is a rephrase of the questions posted regarding measure theory but also including integration. (Reference book on measure theory)

I have to review the knowledge of measure and integration theories for the sake of continue the work that I started in my thesis.

I am not totally new with axiomatic measure theory and integration theory, I have some knowledge due to self-study. Now I feel that all the knowledge has gone and still don’t have a clear overview on the structure of measure and integration theories. And here come my specified requirements for a reference book.

I wish the book elaborates the proofs, since I will read it on my own again, sadly. And this is the most important criterion for the book.

I wish the book covers most of the topics in measure and integration theories. I do want to review both theories at a more general level. If such a condition cannot be achieved, I'd like to more focus on integration.

I wish the book could deal with convergences and uniform integrability carefully.

My expectation is after thorough reading, I could have strong background on measure and integration at an analytic level.

Sorry for such a tedious question.

Answer 1

I am not totally sure what you are looking for, but I would suggest taking a look at Dudley's book:

• Dudley: Real Analysis and Probability

He certainly includes detailed proofs, and covers many important topics in measure and integration (as well as probability). I certainly learned a lot from it, and it has very detailed references at the end of each chapter.

Answer 2

One of the very best books on analysis, which also contains so much more then just measure and integration theory,is also available very cheap from Dover Books: General Theory Of Functions And Integration by Angus Taylor. You can probably get a used copy for 2 bucks or less and it contains everything you ever wanted to know about not only measure and integration theory, but point set topology on Euclidean spaces. It also has some of the best exercises I've ever seen and all come with fantastic hints. This is my favorite book on analysis and I think you'll find it immensely helpful for not only integration theory, but a whole lot more.If you're suffering through Rudin's Real And Complex Analysis,that book will be made much clearer with Taylor as a supplement as it covers in much more details matters Rudin just brushes over.

There are other books,of course,but most of them are pretty expensive and half of them aren't anywhere near as accessible. I personally think this is all the book you need on graduate real analysis.