How to define Borel-measurability of a real-valued functional on separable Hilbert spaces?

Question

Let $H$ be a real separable Hilbert space and let $F : H \to \mathbb{R}$ be a functional.

Then, how do we define Borel-measurability of this $F$? Is it just that $F^{-1}(A)$ is a Borel set in $H$ if $A$ is a Borel set in $\mathbb{R}$?

Also, let $B_r \subset H$ be an open ball of radius $r>0$ at the origin. If $F \mid_{B_r}$ is continuous and $F \mid_{H-B_r} =0$ identically, then I guess such a functional $F$ is clearly Borel-measurable?

Could anyone please clarify for me?