Let $E \subset \mathbb{R}$ a null-set. Show that the subset $ \{(x,y) \in \mathbb{R}^2:x-y \in E \}$ is measurable

Question

Let $E \subset \mathbb{R}$ a null-set. Show that the subset $ \{(x,y) \in \mathbb{R}^2:x-y \in E \}$ is measurable. We already know that if E is a $G_\delta $ set the statement is true. and we want to use the following corollay: Let E $\subset \mathbb{R}^n$ a set, then it exists H $\subset \mathbb{R}^n$ a such that E $\subset H$ and $\lambda^* (E)= \lambda^*(H)$.

Answer 1

Consider the linear map $L:\Bbb R^2\to\Bbb R^2$, $L(x,y)=(x+y,y)$. Then, your set $S$ is $L(E\times \Bbb R)=\bigcup_{y\in\Bbb R}(E+y)\times\{y\}$. Since $S$ is image of a Lebesgue-measurable set by a Lipschitz-continuous homeomorphism, it is measurable as well.