For$ f$ non-negative function, show that $f$ is meseurable if for some $p>0$ $f^p$ is measurable.

Question

For$ f$ non-negative function ($f \geq 0$), show that $f$ is meseurable if for some $p>0$ the function $f^p$ is measurable.

Answer 1

The composition of measurable functions is measurable. We know $g(x) = x^{1/p}$ is measurable. so if $f^p$ is measurable then $f = g \circ f^p$ is also.