Measure theory and Borel sets

Question

Let $f:\mathbb{R}\to\mathbb{R}$ borelian function and $\mu$ borelian measure on $\mathbb{R}$ $\sigma$-finite

1. Given $E\in\mathcal{B}_{\mathbb{R}}$ prove that $D=\{(x,y)\in\mathbb{R}^2:y\in E+f(x)\}$ is a borelian set of $\mathbb{R}^2$.
2. Prove that $x\to\mu(E+f(x))$ is a Borel-measurable function.

SOLUTION

1. Consider the function $\phi(x,y)=y-f(x)$, then the previous function is a Borel function because is difference of two Borel functions. Then we have that for $E\in\mathcal{B}_{\mathbb{R}}$: \begin{equation} \phi^{-1}(E)=\{(x,y)\in\mathbb{R}^2:\phi(x,y)\in E\}=\{(x,y)\in\mathbb{R}^2:y-f(x)\in E\}=\{(x,y)\in\mathbb{R}^2:y\in E+f(x)\}=D \end{equation} and then this last set $D\in\mathcal{B}_{\mathbb{R}^2}$ because the function $\phi$ is Borel measurable.
2. Can someone help me with some hints?

Is point one correct?