## Measurable set has nice property.

Let be a measurable set such that (in the Lebesgue measure). Prove that there exist such that .

*MathOverflow*

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Categories: Combinatorics, Measure Theory, Problem Solving
measure

S is Lebesgue measurable iff for all e>0 exist a compact K_e subset of S that m(S-K_e)<e. We can be sure to a small enough 2-cube in S. Three of the vertex of this 2-cube are the points we want to find. My speak is very bad, sorry. Is the proof correct?

If you notice, one of the vertices of the rectangle should be on the diagonal, so I’m afraid its more complicated than that.

I think I have a solution. Are you still interested or has it been solved by now ?

If you have a solution you are welcome to post it. 🙂