Necessary, but not sufficient for showing $f$ is measurable.

Question

I am trying show that if $\{ x\in [a, b] | f(x) =c\} $ is measurable for all $ c\in \mathbb{R} $ is not sufficient to show that $f$ is measurable on $[a, b] $. \ Possible approach : I thought If I defined a non measurable set and define $ f$ to evaluate whether or not $ x $ is in the set.

Answer 1

I think your idea is good. Take a non-measurable set $A\subseteq [a,b]$ and define $f:[a,b] \to \mathbb{R}$ such that $f(x)=x$ if $x$ is in $A$ and $f(x)=x+b-a+1$ if $x$ is in $[a,b]\setminus A$. Then $f$ is injective so for every $c$ in $\mathbb{R}$, we have that $f^{-1}(\{c\})$ is empty or a point and in particular measurable. But $f^{-1}([a,b])=A$ isn't measurable so $f$ is not measurable.