For what values of $n$ does the measurability of $(f(x))^n$ implies the measurability of $f(x)$?

Question

For what values of $n$ does the measurability of $(f(x))^n$ implies the measurability of $f(x)$?

Is there a general relationship between the measurability this two functions?

Answer 1

If $n$ is odd the $f$ is the composition of $f^{n}$ with the continuous function $x \to x^{1/n}$ so it is measurable. If $n$ is even take a non measurable set $A$ and define $f(x)=1$ if $x \in A$, $-1$ otherwise. Then $f^{n}$ is measurable but $f$ is not.