For what measurable set $A \subset \Bbb R^2$ and Lipschitz function $f$ is $f(A) \subset \Bbb R$ not measurable?

Question

Find a Lebesgue measurable set $A \subset \Bbb R^2$ and a Lipschitz function $f: \Bbb R^2 \to \Bbb R$ s.t. $f(A)$ is not Lebesgue measurable.

I want to find an $A$ and an $f$ such that $f(A)$ is a Vitali set. If we let $x \sim y : \iff x-y \in \Bbb Q$ and $E_x:=\{y \in \Bbb R: x \sim y\}$ then for $g:\{E_x:x\in \Bbb R\} \to \Bbb R \; ; g(E_x)=x$ we have that the image of $g$ is not not measurable.

But how can I find the set $A$ and the function $f$?

Answer 1

Take $f(x,y)=x$ where $A$ is the product of a nonmeasurable subset of the line say $V$, and a singleton. Check that $A$ is measurable and then note $f(A)=V$.