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Let $A,B$ be measurable subsets of $[0,1]$ where $m(A)=m(B)=1/4$. Prove that there exists a $t\in\mathbb{R}$ such that $m(A\cap B_{t})>1/1000$. $B_{t}$ denotes the translation of $B$ by $t$.

Perhaps some kind of set approximation works? as measurable sets can be approximated by "simple" sets (finite unions of intervals)

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    The main issue is to obtain an upper bound of $m(A \cup B_t)$ with some distance from $m(A)+m(B)$. Now $A$ is approximately an interval $I$ (reads $A \subset I \cup E$, where $E$ is a set of small measure here), and $B$ is approximately an interval $J$. Then $A \cup B_t$ is approximately $I \cup J_t$, and by a suitable $t$ you can imagine that $I$ and $J_t$ can have significiant amount of overlap so that $m(I \cup J_t)$ is significantly smaller than $m(A)+m(B)$.2011-08-07
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    By interval I meant to say the union of finitely many open intervals.2011-08-07
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    Soarer:Are you assuming A,B are connected? I think if they were, the solution would be immediate. I think the approximation may work if you do it component-wise; take, e.g., A=[0,1/8) and B=(7/8,1]2011-08-08
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    Em, I did not. Is there a place that I used it implicitly?2011-08-08
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    Well, if A is disconnected, how would you approximate it by an interval? Do you mean a collection of intervals? If A were the union $A_:=[0,1/8)\cup (7/8,1]$ , how could you approximate it by an interval?2011-08-08

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