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I have just moved to the differentiation chapter in Royden's book on real analysis. Unfortunately, many of these proofs are difficult for me to understand, and no matter how many times I read them, they do not make sense, specifically the section on Vitali covers. I'm sure something like this happens to many people in various subjects, so what do you do in this situation so that you can make yourself understand what's going on in these proofs.

user1876508
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    Normally I will look up the proof in another book. –  Nov 12 '13 at 02:44
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    You read on, treating the theorem as a black box and come back to it later. Alternatively, you ask your professor to give some intuition (And btw, Vitali's theorem is notorious for being rather difficult to parse, so don't feel too bad :) – Prahlad Vaidyanathan Nov 12 '13 at 02:44
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    I look at multiple texts when I get stuck. Often you will run into cases where an author proves a result in a very difficult manner and it isn't really necessary. We spent half a lecture proving some result in my measure theory class which took all but a short paragraph in another text to prove using slightly different logic and machinery. – Cameron Williams Nov 12 '13 at 03:17
  • A tactic that I find helps sometimes is: try to think about how I would prove a theorem if I didn't have the textbook. Almost always I realise that I have no hope of proving it, but at the very least, it makes me familiar with the objects in the proof and how they could possibly fit together. – Moss Nov 12 '13 at 03:47

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You should look up the Hilbert Transform and the Hardy-Littlewood Maximal function. This makes the differentiation of the integral to work. These theorems are very hard.

http://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function

ncmathsadist
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When I get stuck on a proof I usually read it a few more times and try to address any gaps I have in my understanding of the concepts in the proof. I look up information related to my gaps either in books or online. I will also look up as many different proofs of the same thing hoping that I come across something that I find more helpful. Sometimes I have no choice but to watch cartoons and sleep on it only to return to it in the morning with a better understanding than I previously had. I have noticed that sometimes I get stuck thinking about something in a certain way and by leaving it alone and returning to it later I can look at it in a different way that I was unable to when I first started.

1233dfv
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