Sum of sets with positive measure contains interval

[hhj5575]

The original problem is as follows:

IF E,F are measurable subset of R
and m(E),m(F)>0
then the set E+F contains interval.

After several hours of thought, I finally arrived at conclusion that

If I can show that m((E+c) \bigcap F) is nonzero for some c in R,
then done.

But such a proposition(actually seems trivial...) is very hard to prove.

Could you give me some ideas for solving it?

 

[mathman]

As you describe it, it doesn't look right. Example: E = all irrationals between 0 and 1,F = all irrationals between 1 and 2. E+F = all irrationals between 0 and 2, it does not contain any interval.

 

[hhj5575]

hmm.. I think E+F = all irrationals between 0 and 2 is not right.

1.9 = pi/4 + (1.9- pi/4)

 

[mathman]

Clarification needed - what is the definition of "+" in E+F. I was assuming you meant union.

 

[morphism]

It's most likely the pointwise sum: $E+F =\{e+f \colon e\in E, f\in F\}.$

The problem in the OP is a well-known (and challenging) one. There's a particularly slick solution if you know a thing or two about convolutions: consider $\chi_E \ast \chi_F.$

 

[Bacle2]

This may help:

http://en.wikipedia.org/wiki/Steinhaus_theorem

 

[Bacle2]

I hope it wasn't too much of a giveaway.