In a course of real analysis I am being asked to prove the following:
Let $(S,\mathcal{A})$ be a measurable space. Let $f:S \rightarrow \mathbb{R} $ be a measurable function and $p \in (0,\infty)$. Show $|f|^{p}$ is measurable.
I know that $|f|$ is measurable and also the product of measurable functions is measurable. So for example if $p=3$, I could write $|f|^3 = |f|\cdot|f|\cdot|f|$ and know that this is measurable. When for example $p=3.24$, how would I deal with $0.24$? Is it true that $|f|^{0.24}$ is also measurable?