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I am a huge fan of Frank Jones's book "Lebesgue Integration on Euclidean Space". It's not as well known as most of the other books mentioned, but I like it for the following reasons.

  1. It is extremely well-written.
  2. On the one hand, it works in $\mathbb{R}^n$ from the offset rather than starting with $\mathbb{R}$. \mathbb{R}$ (though really this is no more difficult, and it is good to train students not to be scared of higher dimensions; also, it makes it a bit easier to draw pictures). However, it is still quite a bit more gentle than most other books, and its perspective is extremely concrete.
  3. It includes a lot of classical material that many books ignore.
  4. Its exercises are fantastic.

I learned the subject from this book back when I was a 2nd year undergraduate (back in 1999!). However, though I now own many other books it is still the one I go back to when I want to remind myself about the basic facts of life about integration theory or measure theory or Fourier analysis.
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I am a huge fan of Frank Jones's book "Lebesgue Integration on Euclidean Space". It's not as well known as most of the other books mentioned, but I like it for the following reasons.

  1. It is extremely well-written.
  2. On the one hand, it works in $\mathbb{R}^n$ from the offset rather than starting with $\mathbb{R}$. However, it is still quite a bit more gentle than most other books, and its perspective is extremely concrete.
  3. It includes a lot of classical material that many books ignore.
  4. Its exercises are fantastic.

I learned the subject from this book back when I was a 2nd year undergraduate (back in 1999!). However, though I now own many other books it is still the one I go back to when I want to remind myself about the basic facts of life about integration theory or measure theory or Fourier analysis.