If $f$ is measurable , prove that $f^n$ is also a measurable function , where n is a positive integer.
My understanding Well if I check $f^1$ is measurable this is obvious since I was already given that f is measurable.
However, Do I have to prove that $f^2$ is measurable and $f^3$ is measurable and so on and then conclude than $f^n$ is measurable. And if so how do I know that for every n that the function $f$ is measurable.What if $f^(100)$ is not measurable per say. Under what circumstances can I conclude that for every n that this is true? I am using the H.L Royden and Walter Rudin but nothing else comes to mind.
Hints:
In general if $u:X\to Y$ and $v:Y\to Z$ are measurable function then composition $v\circ u:X\to Z$ is a measurable function.
Function $g:\mathbb R\to\mathbb R$ prescribed by $x\mapsto x^n$ is continuous, hence measurable if domain and codomain are both equipped with the sigma algebra of Borel sets.