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This is a problem on an old prelim exam in analysis that I'm studying to prepare for my own prelim.

We're to prove that if $E\subseteq[a,b]$ is a Borel set with positive Lebesgue measure, then there exist $x,y\in E,\ x\neq y$ such that $x-y\in\mathbb Q$.

In class we proved that a Vitali set can be uncountable and yet not Lebesgue measurable, and I'm sure that could be useful here, but I can't quite see how. I have showed that $E\oplus q_n\cap E\oplus q_m=\varnothing\ \forall n\neq m$, when $\{q_n\}=\mathbb Q$, because I thought I could make some headway using that fact, but I can't see what to do with it.

Asaf Karagila
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Ceph
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1 Answers1

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If $x-y\notin\Bbb Q$ whenever $x,y\in E$ and $x\ne y$, then the sets $E+q$ for $q\in\Bbb Q$ are pairwise disjoint, and $m(E+q)=m(E)$ for each $q\in\Bbb Q$. Let $Q=\Bbb Q\cap[0,1]$. Then on the one hand

$$0<m(E+Q)\le b-a+1\;,$$

since $E+Q\subseteq[a,b+1]$, and on the other hand

$$m(E+Q)=\sum_{q\in Q}m(E+q)=\sum_{q\in Q}m(E)=\infty\;.$$

Brian M. Scott
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