Let$ J ⊂ R$ be countable and let $A ⊂ R $ have Lebesgue measure zero. Prove that there exists $c ∈ [0, 1]$ such that $A ∩ (c + J) = ∅,$ where $c + J = {c + x : x ∈ J}.
As $A ⊂ R $ have Lebesgue measure zero do not implies $A$ is also countable subset of $R $. I could not find right way to tackle this problem
$J$ is countable, hence we may assume its elements are $q_1,q_2,q_3,\ldots$ The set $$ B=\bigcup_{k\geq 1}(A-q_k) $$ is a countable union of sets with measure zero, hence $\mu(B)=0$ and there is some $c\in[0,1]$ such that $c\in\mathbb{R}\setminus B$. In particular, $c+q_k$ does not belong to $A$ for every $k\geq 1$, hence $c+J$ and $A$ are disjoint sets as wanted.