Construct a (non-measurable) function $f: R\to R$ with the following property

Question

any function $g:R \to R$ such that $|g(x)-f(x)|<1$ for all $x \in R$ is non-measurable. I am thinking about the function $f$ being $f=2\chi_N$, $N$ is a non-measurable set in [0,1]. Thus, $m(x\in R, 1<g(x)<3)=N$. I did it this way in the exam, but I am not sure whether it is right or not...If it is wrong, could you please construct an example function? Thank you!

Answer 1

Your construction is correct. Let $U=\{x\in \mathbb{R}|g(x)>1\}$. $U$ is equal to $N$ since if $x\in N$, then by definition $f(x)=2$, so since $|2-g(x)|<1$ we have $g(x)>1$, and if $x\notin N$ then $f(x)=0$, so $|g(x)-0|<1$ so $g(x)<1$. $U$ is the inverse image of the measurable set $(1,\infty)$ under $g$, so $g$ is not measurable.