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I was doing some analysis when it occurred to me that if I knew the answer to the following question many of the exercises would be simpler. However, I haven't been able to make much headway. Any help will be appreciated.

Question: If I have a set $S \subset \mathbb{R}^n$ such that $S$ is not of measure 0, is it always true that there exists some closed rectangle $C$ such that $C \subseteq S$?

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  • $\begingroup$ The answer is no, $\mathbb{R}-\mathbb{Q}$ being an example (as indicated in the answers). But there are some important instances where the answer is true that are helpful to keep in mind. For instance, the set on which a continuous function is positive (or negative) has positive measure, being that it contains a $k$-cell (rectangle). $\endgroup$ – Sargera Jun 19 '13 at 23:31
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No. Think of $\mathbb R \backslash \mathbb Q$.

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    $\begingroup$ Can't believe I missed this one. Thanks. $\endgroup$ – providence Jun 19 '13 at 23:23
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Counterexample: $[0,1]-\mathbb{Q}$ has a non zero measure, yet it does not contain an interval.

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    $\begingroup$ Doesn't it contain $]1/3, 2/3[$? $\endgroup$ – Patrick Da Silva Jun 19 '13 at 23:20
  • $\begingroup$ You are certainly right $\endgroup$ – Amr Jun 19 '13 at 23:21
  • $\begingroup$ I think my example is better. $\endgroup$ – Patrick Da Silva Jun 19 '13 at 23:21
  • $\begingroup$ @PatrickDaSilva Your now corrected answer looks OK. $\endgroup$ – Pedro Tamaroff Jun 19 '13 at 23:21
  • $\begingroup$ @Peter : I read "measure zero" instead of "non-zero measure" the first time, hence the wrong answer. Now it's fine. $\endgroup$ – Patrick Da Silva Jun 19 '13 at 23:22

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