real analysis - Prove that $f$ is Borel measurable.

Let $\mu$ be a finite Borel measure on $\mathbb R$, i.e. a finite measure on the Borel $\sigma$-algebra $S (\mathbb R)$,
and let $B$ be a Borel subset of $\mathbb R$. Define the function $f$ on $\mathbb R$ by $f (x) =\mu (B + x)$.

(a) Show that $f$ is Borel measurable.
(b) Show that $\int f (x) d\lambda (x) =\int f (x) dx =\mu(R)\lambda(B)$, where $\lambda$ denotes the Lebesgue measure

I am not able to prove that $f$ is Borel measurable. I tried to prove that $f^{-1}(a,\infty)$ is a Borel set but couldn't prove it.