Im in my last year of undergrad and im currently taking a class on measure and integration. We are using Real Analysis by Folland and i'm finding it extremely difficult to follow along in class. Last term i took a course on real analysis and we covered topics like completeness, compactness, banach spaces, stone-weistress and etc and i aced that course. But this term, i have no idea whats going on in lectures and im getting a little worried im trying really hard to study outside of class and im getting nowhere. Is this normal for a senior analysis course? Folland dives straight into the 'abstract' approach to the Lebesgue integral.

I think this is a tough way to be introduced to the Lebesgue integral. All the new definitions seem so weird, and it's hard to understand why we care about them. I recommend looking at a book like Weeden and Zygmund, which takes a more concrete approach to Lebesgue integration, working in R n. (At the end of the book they finally develop the abstract approach.) This may help to motivate the material. Royden might also be a good book to look. Confusion is normal in this class, because the Lebesgue integral is one of the first 'big machine' theories many students see.

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The best way to deal with it is to look at a wide variety of books and try to find two or three that make sense to you, so that for any one topic that confuses you, you can perhaps find one really good explanation between them. Here's a list of some good books, arranged roughly in order from more concrete to more abstract: • Bartle, The elements of integration and Lebesgue measure. I haven't read this but many people recommend it for beginning students. • Stromberg, Introduction to classical real analysis.

A careful treatment of the Lebesgue integral in Euclidean spaces is in chapter 6. • Stein & Shakarchi, Real analysis (Princeton lectures on analysis, volume 3). Stein is a master expositor and this series is full of wonderful concrete examples and exercises but sometimes skimps on the precise statement and proof of the more technical theorems. • Adams & Guillemin, Measure theory and probability. This is a neat little book but has two traits you may not want -- one is the probability flavor to the applications, and the other is that it uses the 'completion of metric space' construction of Lebesgue measure, which is fun but in my opinion a bit obtuse.

• Kolmogorov & Fomin, Introductory real analysis. I first learned integration from this little book, with its oddly conversational Russian exposition, its annoying habit of printing the solutions to exercises directly below the exercises, and its $12 price tag. • Royden, Real analysis. I personally dislike this book for reasons which are unlikely to matter to anyone else. Many people have learned integration from it over the years. Majalah Apo Free Download here. • Rudin, Real and complex analysis.

If you are having trouble with Folland's book, Rudin may not help you much right now, but it's the classic for very good reasons. Give it a try; you might be surprised. In particular you mention that you're having trouble with product sigma-algebras. Integration theory has a number of technical constructions in it that you actually don't care about, except to verify that 'it is what it is supposed to be'. The product sigma-algebra is one of those. You just have to do some technical crapola to verify that your construction of 'product sigma-algebra' produces the right thing (product of the Borel sigma-algebra on R with itself produces the Borel sigma-algebra on R 2, etc.) Then you forget about the whole thing.

I'm a computer science PhD student with no pure math background. I struggled to teach myself measure theory from books, then got it fairly well when I took a class.

You might find the class notes helpful: They're in French but it's not too hard to read math in other languages and google translate will get you pretty far. I am not really a French speaker either. The reason I liked these notes is that the first few chapters illustrate exactly what problems measure theory is trying to get around and they demonstrate how things work when just using the Lebesgue measure on the one dimensional real case. They also have a lot of cool examples. I think a lot of the books dive straight into abstract measure theory before you get an intuition for why any of the abstract machinery needs to be developed. I took a class using this book last semester, and it is hard.

Folland doesn't give you a whole lot to work with in the way of motivation and it can be hard to understand what things are 'supposed' to be. You will eventually realize that the product sigma algebra is just a specific example of a more general construction (in chapter 2.1 I believe.) and it will click. For me, these 'clicking' moments didn't always come from Folland, I had to use other resources to figure out what things 'really' mean and then relating that to what Folland says. I used Royden for this, because it's more concrete and you have intuition for the real line already.

I don't like Rudin for this because his exposition is drastically different; I don't have experience with other measure books. I hated Folland at first but after I realized how to read it I realized that it's a great book. Everything is there, you just gotta figure out a way to find it. PS, the product sigma algebra is exactly what it should be, except in the case where you have uncountably many factors.

Consider that example and see why you need the definition as given, and it'll become perfectly clear.

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