Measure of the set $\{(x+f(x), x-f(x)):x\in \mathbb R\}$ in $\mathbb R^2$ is 0

Question

Let $f:\mathbf R\to \mathbf R$ be a Lebesgue measurable function. Then the set $S=\{(x+f(x), x-f(x)):x\in \mathbf R\}$ is Lebesgue measurable in $\mathbf R^2$ and its measure is $0$.

I am completely stumped here. I think we need to use Tonelli or Fubini's theorem intelligently once we establish that the set $S$ is measurable.

Thanks.

Answer 1

The linear map $(x, y) \mapsto (x + y, x - y)$ sends the graph of $f$ onto your set. The claim follows [Why?].