Let $E\subset\mathbb{R}$ be a measurable set, and define $F=\{(x,y):x-y\in E\}$. Is $F$ a measurable set?

Question

Let $E\subset\mathbb{R}$ be a measurable set, and define $F=\{(x,y):x-y\in E\}$.

Is $F$ a measurable set? why?

Thanks to any one who give me a hint!

Answer 1

By a linear change of coordinates of $\mathbb R^2$ via $(x+y)/2$ and $(x-y)/2$, the map $(x,y)\mapsto x-y$ becomes $(X,Y)\mapsto Y$. (And linear changes of coordinates does not change the measurability of a set.)

Or, equivalently, note that up to scaling by factor of $2$, the map $x-y$ is the projection to the line $y=-x$.

For these maps, that are linear projections, it is clear that inverse image of measurable sets are measurable, because the inverse image of a set $E$ is $E\times \mathbb R$.