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This is a question on a self-study matter. Earlier, I stopped reading W. Rudin's book on "principles of mathematical analysis", at the chapter on Riemann integration. Now for certain other needs, I need to study Lebesgue integration; nothing too difficult; I would imagine the last chapter of this book would suffice without learning the whole difficult machinery as in Rudin's bigger book "Real and Complex Analysis". The purpose is to read a book on ergodic theory later.

Up to now I had read the book line by line, page by page, and right now I am on the chapter on sequences and series of functions. My question is, whether this topic is relevant to Lebesgue integration and if I can skip this. For the earlier chapters, I found that skipping something in between for a topic so basic as analysis is a bad idea and I would miss something. On the other hand, learning a chapter which on the surface has nothing to do with integration, gives a feeling of lack of motivation, and therefore there is less enthusiasm to study. If I can skip this chapter and the rest and go straight to the last chapter, I would perhaps be able to proceed faster, and with more interest.

So, please advice me from the viewpoint of more experienced people, whether I can safely skip the chapter on sequences and series without too much harm.

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My opinion is: no, you can't skip that stuff without impairing seriously your understanding of the subject. If you wanted to cover the subject just for an urgent exam or something like that then perhaps you can skip that stuff and just check here and there whenever you need within Lebesgue Integration...but I still think it'd be hard for you to follow. –  DonAntonio Jun 9 '12 at 14:42
    
Who downvoted the question and why, if may I ask? –  DonAntonio Jun 9 '12 at 14:43
1  
@Don: No votes from me, but what is the question here, anyway?? (Or maybe more to the point, could we just add CW answers "yes, skip" and "no, don't skip" ?) –  The Chaz 2.0 Jun 9 '12 at 14:45
    
@TheChaz The question is the last lines in the OP, and he's talking of studying Lebesgue Integration without going through sequences and series. –  DonAntonio Jun 9 '12 at 14:47
    
Sequences of functions are essential for USING Lebesgue integration, and sometimes even for defining Lebesgue integration (depending on the approach). Lebesgue integration to Riemann integration is like the real numbers to the rational numbers. You need the real numbers to appropriately handle situations when sequences and series of numbers arise, and in an analogous way, you need Lebesgue integration to appropriately handle situations when sequences and series of functions arise. –  Dave L. Renfro Jul 17 '12 at 13:55

1 Answer 1

You can skip on first reading, and go straight to the chapter on Lebesgue integration you are interested in. Then if and when you encounter something that involves sequences or series of functions which needs clarification, you can go back and read the relevant chapter piece by piece, or as a whole, depending on your interest. My guess is by that time you will develop a will to read this chapter on sequences or series of functions since you would have encountered many motivating reasons. On another, more general note, you should learn about sequences and series of functions sooner or later because their importance is not limited to the context of Lebesgue integration.

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Can you explain why -1? I am not trolling. I spent some effort answering this question. If you have an argument to support the view that this answer is downright incorrect, please provide here, since it would serve the reader better than just a downvote. –  timur Jun 9 '12 at 14:58

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