How to check that a random variable is in $L^p(\Omega,F,\mathbb{P})$?
Maybe I should provide an example to get my point across.
Let $X_\alpha$ be a random variable with CFD $F_\alpha(x) = (1 - \frac{1}{x^\alpha})\mathbb{1}_{[1,\infty]}(x)$
I want to find for which values $\alpha$ $X_\alpha$ belongs to $L^p$. Should I look at integrating the density function to the power p in absolute value or rather the CFD? I am a bit confused.
$X_\alpha$ is nonnegative, so it suffices to check $E X_\alpha^p < \infty$.
Since you have the CDF, it is easiest to use the tail sum probability for this expectation.
$$E X_\alpha^p = \int_0^\infty P(X_\alpha^p > t) \, dt = \int_0^\infty (1 - F_\alpha(t^{1/p})) \, dt.$$
Hopefully this helps you finish the question.