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Let $A,B\subset \mathbb{R}^n$ be Lebesgue measurable sets and assume that for every $x\in \mathbb{Q}^n$, there exists a null set $N_x$ such that $A+x\subset B\cup N_x$. Show that if $A$ is not a null set, then the complement of B in $\mathbb{R}^n$ is a null set.

I have spent a fair amount of time on this question, and can't seem to get anywhere. I started by noting that if we take the union of the $N_x$, to obtain a set $N$, this set is still measure $0$. So $A+x\subset B\cup N$. Then, if $B_1=\cup_{x\in \mathbb{Q}^n}A+x$, we have $B_1\subset B\cup N$. Then I assume that $A$ has positive measure, so that $B_1$ does. And also, $B_1$ is invariant under rational translations. if $B_1^c$ has positive measure, it must also be invariant under rational translations. I have no idea where to go from here, or even if this is the correct direction.

Kevin Johnston
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  • This result might seem mildly related: https://math.stackexchange.com/questions/103306/for-a-set-of-positive-measure-there-is-an-interval-in-which-its-density-is-high ...? – user357980 Jun 20 '17 at 06:57
  • I got an idea. If $U \subset \mathbb R^n$ is open and $E \subset U$ is measurable with $m(E) < m(U)$, does there exist an open set (or ball) $V \subset U \setminus E$? If so, then let $U$ be some open set in $\mathbb R^n$ and $E = (B \cup N) \cap U$. Then, if $U \setminus E$ is not a null set, there exists open $V \in U \setminus E$. But an open set must contain a subset of $A + \mathbb Q^n$, contradicting to $A + \mathbb Q^n \subset B \cup N$. – md2perpe Jun 21 '17 at 10:44
  • Unfortunately it's not true that there must exist an open set $V \subset U \setminus E$. As an example, let $(q_k){k=1}^{\infty}$ be an enumeration of $\mathbb Q$ and $U = \mathbb R$ and $E = \bigcup{k=1}^{\infty} (q_k - \frac{1}{k^2}, q_k + \frac{1}{k^2})$. Then $m(E) < \infty$ but $E$ is dense so $U \setminus E$ contains no open balls. – md2perpe Jun 21 '17 at 15:16
  • Have you had Lebesgue's density theorem? https://en.wikipedia.org/wiki/Lebesgue%27s_density_theorem – md2perpe Jun 21 '17 at 15:47

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