Let $μ$ be the Lebesgue measure in $\mathbb{R}$ and let $A\subset \mathbb{R}$ be a borel set such that $μ(A)> 1$. I want to show that there exist $x,y \in A$ such that $x-y \in \mathbb{Z} -\{0\} $. If $B\subset \mathbb{R}$, we define $$B+x=\{b+x : b\in B\}$$
I need to follow these steps:
a) Show that $$μ(A)=\sum_{n \in \mathbb{Z}} μ((A-n)\cap P)$$ where $P=[0,1)$
b) Show that the sets $(A-n)\cap P$ are not disjoint, and so, the sets $(A-n)$ are not disjoint. Conclude.
I already know that the Lebesgue measure is translation invariant but I don't know how to start with part a). Please help me with this.
I believe part b) follows using part a) since $((A-n)\cap P)\subset P$ so if they were disjoint, by the σ-additivity of dijoint sets and monotony we would have that $$μ(A)=\sum_{n \in \mathbb{Z}} μ((A-n)\cap P)= μ(\cup (A-n)\cap P ) \leq μ(P)=1$$ which contradicts the hypothesis that $μ(A)> 1$.
I need help with the first part so I would appreciate very much yours! Thank you!!
HINT:
$P+n$ form a partition of $\mathbb{R}$, so $A\cap (P+n)$ form a partition of $A$. Now use $\mu(A\cap (P+n) = \mu((A-n) \cap P)$