Sum of sets with positive measure contains interval

In summary, the conversation discusses a problem involving measurable subsets of the real numbers and their sum. After hours of thought, it is concluded that showing the intersection of two subsets is non-zero can solve the problem. However, this proposition is difficult to prove and the speaker asks for help in finding a solution. The conversation also mentions a potential solution using convolutions.
  • #1
hhj5575
8
0
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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  • #2
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.
 
  • #3
hmm.. I think E+F = all irrationals between 0 and 2 is not right.

1.9 = pi/4 + (1.9- pi/4)
 
  • #4
Clarification needed - what is the definition of "+" in E+F. I was assuming you meant union.
 
  • #5
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
I hope it wasn't too much of a giveaway.
 

Related to Sum of sets with positive measure contains interval

1. What is the definition of a "sum of sets with positive measure"?

The sum of sets with positive measure refers to the union of two or more sets that each have a positive measure, or non-zero length. In other words, the combined total of the sets' measures is greater than zero.

2. How is the sum of sets with positive measure related to intervals?

The sum of sets with positive measure can contain intervals because an interval is a set with a positive measure. When two or more sets with positive measure are combined, the resulting sum may contain intervals as well.

3. Can the sum of sets with positive measure contain any type of interval?

Yes, the sum of sets with positive measure can contain any type of interval, including open, closed, half-open, or half-closed intervals. As long as the interval has a positive measure, it can be included in the sum of sets.

4. How is the sum of sets with positive measure different from the sum of sets with zero measure?

The main difference between the sum of sets with positive measure and the sum of sets with zero measure is that the former results in a set with a positive measure, while the latter results in a set with zero measure. This means that the sum of sets with positive measure can contain intervals, while the sum of sets with zero measure cannot.

5. What is the significance of the "sum of sets with positive measure contains interval" concept in mathematics?

This concept is important in various areas of mathematics, including measure theory, set theory, and real analysis. It helps to understand the structure and properties of sets with positive measure, and how they can be combined to form larger sets. It also has applications in probability and statistics, as well as in other branches of science and engineering.

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