Prove a set has measure 0

Question

Let $A\subset\mathbb{R}$ be a set has measure $0$. Prove that the set $\{(x,y)\in\mathbb{R}^2: \,x-y\in A\}$ also has measure zero.

I don't know how to start. Please help me or give me a hint. Thank you.

Answer 1

Hint: let $B:=\left\{(x,y)\in\mathbb{R}^2: \,x-y\in A\right\}$ and $B_x:=\left\{y\in\mathbb{R} : \,x-y\in A\right\}$. By Fubini's theorem, $$\lambda_2\left(B\right)=\int_{\mathbb R}\lambda\left(B_x\right)\mathrm dx.$$ What is $\lambda\left(B_x\right)$?