Measurability of $f(x+x_0)$ for a measurable function $f$

Let $f$ be a (Lebesgue) measurable function defined on $\mathbb{R}^n$. Given a vector $x_0$ in $\mathbb{R}^n$, I would like to know whether the function $f(x+x_0)$ is measurable or not. I know $\Phi\circ g$ is measurable whenever $\Phi$ is continuous and $g$ is measurable, and a book warns me of an example of a measurable function $g$ and a continuous function $\Phi$ such that $g\circ\Phi$ is not measurable. However, I have no idea how to prove or disprove measurability of $f(x+x_0)$. Can someone please give me a hand? Thank you very much.

Let $g(x) = x + x_0$. $E := f^{-1}((a, \infty))$ is a measurable set by definition, and $g^{-1}(E) = \{x - x_0 : x \in E\}$, i.e. $E - x_0$. But the translation of a Lebesgue measurable set is Lebesgue measurable, so $f \circ g$ is measurable.