Let $X$ be a non-negative random variable on $(\Omega, \mathcal{F}, P)$ such that $E[X]<\infty$. I need to show that:
$E[X]=\int_0^\infty P(X>t) \ dt$ (Riemann integral)
where $E[X]=\int_\Omega X \ dP =\int_\mathbb{R}x \ dP_X$ (Lebesgue integral).
Any help would be appreciated, thank you.
\begin{align} \mathbb E[X]&=\int_\Omega X\,\mathrm d \mathbb P\\ &=\int_\Omega \int_0^\infty \boldsymbol 1_{[0,X]}(t)\,\mathrm d t\,\mathrm d \mathbb P\\ &\underset{\text{Fubini}}{=}\int_0^\infty \int_\Omega \boldsymbol 1_{[0,X]}(t)\,\mathrm d \mathbb P\,\mathrm d t\\ &=\int_0^\infty \mathbb P\{X>t\}\,\mathrm d t. \end{align}
You can make more general by proving that if $\varphi \in \mathcal C^1(\mathbb R)$, s.t. $\varphi \geq 0$ and $\varphi (0)=0$, then $$\mathbb E[\varphi (X)]=\int_0^\infty \varphi '(t)\mathbb P\{X>t\}\,\mathrm d t.$$
