Sum of sets with positive measure contains interval

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Discussion Overview

The discussion revolves around the mathematical problem concerning the sum of measurable sets with positive measure in the real numbers. Participants explore whether the sum of two measurable sets, each with positive measure, necessarily contains an interval. The conversation includes theoretical considerations and challenges related to proving this proposition.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that if the measure of the intersection of the translated set and another set is nonzero, then the sum contains an interval.
  • Another participant challenges this by providing a counterexample involving sets of irrationals, suggesting that their sum does not contain any interval.
  • A third participant questions the definition of the sum operation used in the original problem, suggesting it might refer to union rather than pointwise addition.
  • It is clarified that the sum is indeed the pointwise sum of the sets, defined as the set of all sums of elements from each set.
  • A later reply mentions a known theorem related to this problem, hinting at a solution involving convolutions.

Areas of Agreement / Disagreement

There is disagreement among participants regarding the validity of the original claim. Some participants provide counterexamples that challenge the assertion that the sum of two measurable sets with positive measure contains an interval, indicating that the discussion remains unresolved.

Contextual Notes

Participants have not reached a consensus on the definitions and implications of the sum of sets, and there are unresolved mathematical steps regarding the proof of the original proposition.

hhj5575
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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?
 
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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.
 
hmm.. I think E+F = all irrationals between 0 and 2 is not right.

1.9 = pi/4 + (1.9- pi/4)
 
Clarification needed - what is the definition of "+" in E+F. I was assuming you meant union.
 
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.##
 
I hope it wasn't too much of a giveaway.
 

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