How to prove the following measure theory result?

Question

Let $A$ and $B$ be Lebesgue measurable subsets of $(0,1)$ such that $m(A)>1/2$ and $m(B)>1/2$. Prove that there exist $a \in A$ and $b \in B$ such that $a+b=1$.
I was doing this by assuming that it doesn't hold then $\forall a \in A$ and $b \in B$ we have either $a+b>1$ or $a+b<1$ then I wasn't able to get some contradiction.

Answer 1

Hint: Consider the set $C=\{1-a:a\in A\}$.

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Our goal is to show that $B\cap C\neq\emptyset$. But $m(A)=m(C)$, since $a\mapsto 1-a$ preserves measures. So $m(B)+m(C)>1$, which implies $B\cap C\neq\emptyset$ since $B$ and $C$ are both contained in $(0,1)$.