Home > Combinatorics, Measure Theory, Problem Solving > Measurable set has nice property.

## Measurable set has nice property.

Let $S \subset [0,1]^2$ be a measurable set such that $m(S)>1/2$ (in the Lebesgue measure). Prove that there exist $x,y,z$ such that $(x,y),(x,z),(y,z) \in S$.

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1. March 4, 2011 at 8:20 pm

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?

2. March 4, 2011 at 8:25 pm

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

3. April 25, 2013 at 11:04 am

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

• April 25, 2013 at 12:03 pm

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