Though unlikely for such a technical topic, is there a book on measure theory (and/or Lebesgue integration) for the casual learner? I am especially interested in the history: the motivation behind the theory, how it came to be, why mathematicians felt they needed some other tool, and how it was developed. Easy examples that would make a self-learner "tick" would be great.

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    $\begingroup$ Measure theory proper or Lebesgue integration? For the latter, there's Bressoud's A Radical Approach to Lebesgue's Theory of Integration which gives plenty of motivation and historical developments (it's still an introductory textbook though, so you'll probably need to work a few exercises along the way to get the most of it) $\endgroup$ – Weaam Aug 4 '17 at 18:32
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    $\begingroup$ @Weaam Thanks for the comment. Edited the post accordingly. $\endgroup$ – blackened Aug 4 '17 at 18:33
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    $\begingroup$ You could look at "A Garden of Integrals" by Franke E. Burk. It is a bit different from what you are asking for, but it does place the Lebesgue integral and measure in the context of the general history of integration. $\endgroup$ – Wraith1995 Aug 4 '17 at 18:48
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    $\begingroup$ Hawkins's Lebesgue's Theory of Integration: Its Origins and Development is nice, but perhaps doesn't qualify as a layman's book. $\endgroup$ – Hans Lundmark Aug 5 '17 at 7:14

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