If $f^4$ is measurable, is $f$ measurable?

Question

If $f^4$ is measurable, is $f$ measurable?

I know that if $g$ is continous on $\mathbb{R}$ and $f$ measurable, then $g\circ f$ is measurable on $\mathbb{R}$.

If $g(x)=x^4$, then $(g\circ f)(x)=f^4(x)$.

I have that $g$ is continous. So $(g\circ f)$ measurable iff $f$ is measurable. Is my conclusion true?

Answer 1

This isn't correct. Let $M$ be a non-measurable set. Then $$f(x)=1\text{ if }x\in M,f(x)=-1\text{ otherwise}$$ then $f$ is not measurable, but $f(x)^4$ is constant $1$, hence measurable.

Answer 2

Let $\mathcal N$ a non measurable set. What do you think about $$f(x)=\begin{cases}1&x\in \mathcal N\\-1&x\in \mathcal N^c\end{cases}\ ?$$