Eden Solomon

2022-06-24

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 $f\left(x+{x}_{0}\right)$ is measurable or not. I know $\mathrm{\Phi }\circ g$ is measurable whenever $\mathrm{\Phi }$ is continuous and $g$ is measurable, and a book warns me of an example of a measurable function $g$ and a continuous function $\mathrm{\Phi }$ such that $g\circ \mathrm{\Phi }$ is not measurable. However, I have no idea how to prove or disprove measurability of $f\left(x+{x}_{0}\right)$. Can someone please give me a hand? Thank you very much.

Quinn Everett

Expert

Let $g\left(x\right)=x+{x}_{0}$. $E:={f}^{-1}\left(\left(a,\mathrm{\infty }\right)\right)$ is a measurable set by definition, and ${g}^{-1}\left(E\right)=\left\{x-{x}_{0}:x\in E\right\}$, i.e. $E-{x}_{0}$. But the translation of a Lebesgue measurable set is Lebesgue measurable, so $f\circ g$ is measurable.

Hailie Blevins

Expert

Thank you, and I guess you have used $\left(f\circ g{\right)}^{-1}\left(\left(a,\mathrm{\infty }\right)\right)=E-{x}_{0}$