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Let $E$ and $F$ be measurable sets with $m(E),m(F)>0$. Prove that $E+F$ contains an interval.

This is a part of an exam preparation, I would appreciate a hint. Thanks!


marked as duplicate by Seirios, Peter Taylor, Micah, Julian Kuelshammer, Lord_Farin Jun 25 '13 at 7:39

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  • 1
    $\begingroup$ A Google search throws up the Steinhaus theorem? $\endgroup$ – copper.hat Jun 21 '13 at 14:56
  • 2
    $\begingroup$ See the answer here. $\endgroup$ – David Mitra Jun 21 '13 at 14:57
  • $\begingroup$ @DavidMitra: Thanks but I'm not familiar with some of the notions there... $\endgroup$ – catch22 Jun 21 '13 at 19:02

Don't look at the answer!!!!!!

Ok, there are at least three ways to do this problem. Here are some hints for each way:

(1) Just beast it out without any advanced measure theory ideas. This is the way I would NOT recommend.

(2) Since your sets $E$ and $F$ have positive measure, they have points $e\in E$ and $f\in F$ where the metric densities of your sets are one at those points. Consider the point $e+f$ and use the fact just stated.

(3) Use the Fourier transform on the function $\chi_{E} \ast \chi_{F}$ and see what you come up with.

Have fun!

  • $\begingroup$ Thanks Alex. About 2 and 3: I'm not familiar with what you said. About 1: Can you be more specific? $\endgroup$ – catch22 Jun 21 '13 at 19:03
  • $\begingroup$ The solution is messy if just try to attack the problem without using any ideas like the Fourier transform or metric density. It can be done (see above) but the solutions given by (2) and (3) are much nicer. $\endgroup$ – Alex Lapanowski Jun 22 '13 at 23:04

Here's an easy way to see this: Consider the function $f(x) = \mu(E \cap (F + x))$. Observe that $f$ is continuous and not identically zero.


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