A "box" is a cartesian product of intervals of the type $[a,b]$

I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite difficult

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I am aware that a lot of posts already exists in this direction, for example this one: Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] But it is always about intervals on $\mathbb{R}$ and the proofs are all fairly tough.

Does anyone know if there exists a reference of this proof that is sufficiently easy for beginners in analysis?


Chapter 1 of Stein and Shakarchi's Real Analysis textbook has a similar result, and the proof includes a helpful picture. But I think it's worth taking the time to understand Tao's proof. Dyadic cubes are quite useful.

  • $\begingroup$ Thanks, I will take a look at it. I'm using this book where literally every proof cites "every open set can be written..." yet I do not know how to prove this. By the way is the proof of this fact considered "natural" proof in the sense it can be proved on the spot and does not require some sort of gimmick? I want to assess whether if this is something that might show up on a test....:-) $\endgroup$
    – Fraïssé
    Apr 26 '16 at 2:27
  • $\begingroup$ I suppose it is, though natural is of course a subjective term. It's important to note that only in $\mathbb{R}$ can the intervals be made disjoint; in higher dimensions "almost disjoint" is the best you can do. $\endgroup$ Apr 26 '16 at 2:29
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    $\begingroup$ In higher dimensions, if you use intervals of the form $(a_1,b_1]\times\cdots(a_n,b_n]$ then they can be made disjoint. This is useful when applying countable additivity of measures. $\endgroup$
    – user169852
    Apr 26 '16 at 16:51
  • $\begingroup$ @Bungo: Those aren't open though. $\endgroup$ Apr 26 '16 at 18:01
  • $\begingroup$ @carmichael561 Of course. I just wanted to point out the "workaround" that can be used if a countable disjoint union of intervals (of some kind) is desired. $\endgroup$
    – user169852
    Apr 26 '16 at 20:10

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