Suppose $f : \mathbb R^n → \mathbb R$ is continuous and $A ⊂ \mathbb R^n$ a compact set. Furthermore, $\mathcal X_A$ is the characteristic function with respect to A, i.e. $$\mathcal X_A(x) = \begin{cases} 1, &\text{ if } x ∈ A \\ 0, & \text{ if } x\notin A \end{cases}$$
Show: The product $\mathcal X_A · f : \mathbb R^n → \mathbb R$ is $B(\mathbb R^n)-B(\mathbb R)$-measurable and integrable.
Can I say $f : \mathbb R^n → \mathbb R, f^{-1}(A)\in \mathcal A$ for all Borel sets $A\subset \mathbb R$? That the multiplication with a measurable map is Borel measurable because all the pre images of Borel algebra are in $\mathcal A$? But other than that I am not sure how to prove it.
If anyone can explain a bit how I am supposed to continue I would be very grateful!