let non-negative $X\in L^p(\Omega,\Sigma,\mathbb{P})$ and from norm definition we have
$$\big\| X\big\|^p_p=\int|X|^pd\mathbb{P}$$
I've seen another expression below called property of the $L^p$ norm for $p\in (1,\infty)$
$$\big\|X\big\|^p_p=p\int^{\infty}_0\lambda^{p-1}\mathbb{P}(X\geq\lambda)d\lambda$$
I am not sure if I understand correctly, my gut have some vague understanding of $\mathbb{E}(|X|^p)=\big\|X\big\|^p$, and the second equation somewhat looks like the introductory definition of $\mathbb{E}(|X|^p)$, but isn't it should be $$\int^\infty_0\lambda^p\mathbb{P}(X\geq\lambda)d\lambda$$
Please enlighten me more about this $\big\|X\big\|^p_p=p\int^{\infty}_0\lambda^{p-1}\mathbb{P}(X\geq\lambda)d\lambda$