Real analysis: measurable function $f$ and non measurable function $g$ such that their composition is measurable

Question

The title says it all.

I need to find an example of a measurable function $f$ and a non measurable function $g$, such that their composition $f\circ g$ is measurable, but I cant think of anything useful. An explanation would also be nice.

Answer 1

Let $A \subseteq \mathbb{R}$ be a non-measurable set. Let $g : \mathbb{R} \to \mathbb{R}$ be defined as:

$$g(x) = \begin{cases} 1, & \text{if $x \in A$} \\ -1, & \text{if $x \notin A$} \end{cases}$$

$g$ is non-measurable because $g^{-1}(\{1\}) = A$, which is non-measurable.

Let $f$ be the absolute value. $f$ is measurable since it is continuous.

Then $f \circ g = 1$, which is measurable.

Answer 2

An uninteresting example: $f=0$ and $g=\chi_{E}$, where $E$ is any non-measurable set, then $f\circ g=0$.