How should I study measure theory?

Question

Im about to start studying measure theory on my own but I dont know what order I should follow, plus I dont know which textbook I should use. Any ideas? I've already had a course on measure theory but it was kinda limited on the Lebesgue measure on the real numbers, integrals, $L^p(\mathbb{R})$ spaces etc; I'd like to study the more general measure theory. Also, any tips or advice for the subject would be appreciated

Answer 1

These are the book's i would propose :

1)Terence Tao's Measure Theory

It is an excellent book and also has a small chapter in which he introduces you to some strategies for tackling problems in this subject.

2)One of my favourites is Stein's Real analysis:Measure Theory,Integration and Hilbert Spaces.

This book begins with an introduction to Lebesgue measure in $\mathbb{R}^n$ and then goes more abstract.And also it has great exercises.

3)Inder.K.Rana's Measure and Integration This book is not known widely but still its a great book with many examples and interesting exercises.

For more advanced book then you have the classic Real and Complex Analysis of Walter Rudin and Folland's Real Analysis:Modern Technics and their Applications

I would suggest for now you to start with one of the first 3 books i mentioned.

I hope my answer helps you.

Answer 2

Just find a book and start reading. As is with all subjects in math, it's best to try to come up with your own proofs before reading the author's, and to also ask yourself questions. If there is some bit that you are struggling with a lot, then check other sources for a different explanation or post a question here for clarification. Do lots of exercises. No one can put it more beautifully than Paul Halmos: $$ `` \text{Don't just read it; fight it.}" $$

The only advice I can think of that is specific to this subject is that constructions really matter and will come back over and over again. In lots of topics, you will construct something, describe it's properties, and then proceed to only use those properties. This is not true of measure and integration theory. Knowing how to extend a premeasure to an outer measure, and then how to obtain a measure from an outer measure will come back when solving problems well beyond the chapter on measures.

Also, when solving problems regarding integration, one will often restrict to simple functions (or another dense class of "nice" functions), then obtain the result for positive functions, then real-valued functions, and finally complex-valued functions. One of my professors has joked that "there is only one measure theory proof" in regards to this method of first solving it for a "nice" function and then using a density or construction argument from there.

I am a student and I haven't had the opportunity to read through many books on measure theory, so I don't feel very comfortable giving a book recommendation. I first learned from Stein and Shakarchi's "Real Analysis: Measure Theory, Integration, and Hilbert Spaces" and later from Folland's "Real Analysis: Modern Techniques and Their Applications." Among the two, I very much preferred Folland - but that could be simply because it was my second exposure. Folland also begins with the abstract treatment, so between the two, I think you would prefer it as well. Be sure to check out other questions regarding book recommendations, such as this one.