measurability of function and its integer power

Question

$f(x)$ is a real valued function and if $(f(x))^3$ is Lebesgue measurable, then prove that $f(x)$ is measurable.

Answer 1

Note that the composition of a measurable functions with a continuous function is itself measurable.

So, if you take $g(x) = x^{1/3}$, which is monotonically increasing and continuous and therefore measurable and compose it with $f(x)^3$, which is measurable by construction you get $$(g\circ f^3)(x) = g(f(x)^3) = f(x)$$

which implies that $f$ is itself measurable