Please recommend a good textbook on measure theory, real analysis

Question

I have read that G.B.Folland's real analysis. I like the contents that I covers because, like baby Rudin, I can study a lot of content fast. Nevertheless, I took a lot of time to understand since the proofs of the book are omitted much. A lot of proofs that (in my thought) would have been okay to be explained more were omitted.

So I am looking for a more descriptive book that covers Folland's contents and its approaches. I have looked through some other books but I don't think I've ever seen a book that covers some topics like Radon-Nikodym theorem or Caratheodory's theorem as detailed as Folland's.

I have heard that 'The way of analysis' is Baby Rudin's manual, and I wonder if there is a book in a similar position about Folland's real analysis book.

Answer 1

I like Bartle's Elements of Integration. It is straightforward and well-written.

Answer 2

I'm not aware of many books which cover the same breadth of topics as Folland while explaining proofs in more detail. Graduate level textbooks expect you to fill in the details of proofs; this is a necessary aspect of reading math.

That being said, Stein and Shakarchi's Real Analysis and Axler's Measure, Integration, and Real Analysis are both very good options for readability; they cover a similar breadth, although Axler's set of topics is a bit more standard. In both cases, you'll have to fill in details of proofs from time to time.