Book about intuition behind Lebesgue measure

Question

I recently completed a course in Real analysis covering Lebesgue and Borel measure, Fourier series, $L^p$ spaces and such. I can solve problems in these topics but am afraid that I do not truly understand these concepts. I have read about intuition of bits and pieces here on MSE. But are there any books on intuition behind such concepts (or atleast measure theory).

The flavor I am looking for is similar to the one found in "Visual Complex Analysis" by Needham. It does not have to be rigorous as long as it includes intuition/physical interpretation of the idea.

Answer 1

Try A Radical Approach to Lebesgue’s Theory of Integration by Bressoud. Read a review.

Answer 2

I'm not sure you are going to find a satisfactory book that will give you "intuition" for measure theory. It is pretty abstract stuff and I know I definitely didn't "get it" the first time I went through it. You best bet is to just keep working through problems and over time, you'll gain the insights you desire. I will also say that being able to admit that you don't feel confident in your level of understanding means that you are likely further along than most of your peers. If you felt fully confident in your knowledge at this point, chances are you have a very superficial understanding of the material.

In my own experience, I found that many of the concepts from measure theory became more obvious to me once I studied functional analysis (which will give you more insight into Fourier analysis and $L^{p}$spaces). I'd recommend Probability and Measure Theory by Robert Ash even if your primary interest isn't probability theory. It gives a good overview of both measure theory and the relevant areas of functional analysis and often dealing with more general measure spaces than $\mathbb{R}^{n}$ can make the concepts more clear since $\mathbb{R}^{n}$ is so familiar (sometimes this leads one to become biased against new perspectives) from undergrad classes.

http://www.amazon.com/Probability-Measure-Theory-Second-Robert/dp/0120652021/ref=sr_1_1?ie=UTF8&qid=1462929511&sr=8-1&keywords=robert+ash