Distance set of a Borel set is a Lebesgue set.

Question

Let $A \subset \mathbb{R}^n$ be a Borel set and define $D(A) := \{|x-y| : x, y \in A \}$. How to show that $D(A)$ is a Lebesgue set? and maybe even possibly a Borel set? The only idea I have right now is to use the fact that Borel set under Lipschitz continuous function is a Lebesgue set, but I'm not sure how to continue. Any help would be appreciated, thanks.