I am having trouble with this question. It is not an homework question, I am currently trying to practise different problems for an exam.
Let $f$ be a nonnegative measurable function on $\mathbb R^d$, with
$m(\{x | f(x)>\lambda\}) = \frac{1}{1+\lambda^2}.$
For which values of $p$ is $f\in L^p$?
Clearly for $p=1$, you have $\int f = \int_0^{\infty} m(\{x | f(x)>\lambda\}) d\lambda < \infty$.
So $f\in L^1$. But how can I use the sets $\{x | f(x)>\lambda\}$ to compute $\int f^p$ for $p>1$?
Any help would be appreciated! Thank you!