Let $A \subset \mathbb{R}^d$ be a Borel set, $ e \in \mathbb{R}^d$ such that $||e|| = 1$ and define $P_e(a) = ea :\mathbb{R}^d \to \mathbb{R}$ the inner product. Show that $P_e(A)$ is Borel.
We know that $P_e$ is continuous and Lipschitz with constant $1$, how does this help us in showing the statement?
