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I would like to teach myself measure theory and I am looking for a good introductory textbook at the advanced undergraduate or early graduate level. A reader-friendly text with plenty of examples would be ideal.

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I find some lectures here: youtube.com/watch?v=SHV2OMQ-QCo This was very useful for me. You have to play with Mod-01 Lec-04, to find other lectures. –  Cortizol Feb 19 '13 at 20:01
    
Thank you! I find online lectures to be very helpful. –  B0112358 Feb 19 '13 at 20:03
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I like Bartle's book "The Elements of Integration and Lebesgue Measure" –  Marra Feb 19 '13 at 20:06
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This question has been asked before. See math.stackexchange.com/q/71418/4583 and math.stackexchange.com/q/46213/4583 –  Ayman Hourieh Feb 19 '13 at 20:40
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4 Answers

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I am a big fan of Bartle's Elements of Integration. You will also want to see a nice development in the special case of Euclidean Space. For this I recommend Wheeden and Zygmund's book, Measure and Integral.

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Bartle is great. It is also sufficiently similar to big Rudin in order to serve as a good preparation for the latter. –  Michael Greinecker Feb 19 '13 at 20:46
    
Bartle takes a compact and manageable path through the subject that renders more advanced treatments (such as Rudin's Real and Complex) more accessible. I think it is important to understand both abstract measure theory as well as Lebesgue integration on Euclidean spaces. –  ncmathsadist Feb 19 '13 at 22:53
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I highly recommend Royden's real analysis book. The latest version has a large errata but is well broken down in bite sized chunks for self learning. I know this from experience. The older editions have a smaller errata but also require u to fill in more details. Others may disagree with me here but the worst book for self learning Measure theory would be the big Rudin. The concepts are not reinforced enough for newbies. You could allways read both. I would suggest work through Royden and then do exercises only from Rudin's book. All the best.

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I would recommend Measures, Integrals and Martingales by Schilling. It starts out very basic with the notion of $\sigma$-algebras, measures and measurable mappings and also covers integration theory including the most commonly used covergence theorems. Further topics such as martingales, Radon-Nikodym's theorem and conditional expectations are also treated.

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Do like young Grothendieck, try to rebuild Lebesgue theory yourself.

If you fail, Bartle is a good read, and Rudin is not bad either.

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