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Sum of sets with positive measure contains interval

  1. Mar 25, 2012 #1
    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) [itex]\bigcap[/itex] 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?
     
  2. jcsd
  3. Mar 25, 2012 #2

    mathman

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    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.
     
  4. Mar 25, 2012 #3
    hmm.. I think E+F = all irrationals between 0 and 2 is not right.

    1.9 = pi/4 + (1.9- pi/4)
     
  5. Mar 26, 2012 #4

    mathman

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    Clarification needed - what is the definition of "+" in E+F. I was assuming you meant union.
     
  6. Mar 27, 2012 #5

    morphism

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    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.##
     
  7. Mar 27, 2012 #6

    Bacle2

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  8. Mar 27, 2012 #7

    Bacle2

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