Rewrite $p$-th moment

Question

Let $X \in L^p(\Omega, \mathcal{A}, P)$ be a non-negative Random variable for some $p \in (1, \infty)$.

We write

$$E[X^p]=E\left[\int_0^{X} p \lambda^{p-1} d \lambda\right]$$

and I have absolutely no idea why one can write it like this. Do I miss something essential?

Answer 1

Let $X$ be a random variable in $\mathscr L^{p} (\Omega, \mathscr A, \mathbb P)$

$$x^p=\int_0^{x} p \lambda^{p-1} d \lambda$$

$$ \to X(\omega)^p=\int_0^{X(\omega)} p \lambda^{p-1} d \lambda \ \forall \omega in \Omega$$

$$ \to E[X^p]=E[\int_0^{X} p \lambda^{p-1} d \lambda]$$

Nonnegativity is not needed, but p-integrability is.