Show that $\mathbb{E}(X) = \int^\infty_0 \mathbb{P}(X\geq x) dx$

Question

I am learning probability theory and I have trouble going from $\int_\Omega X d \mathbb{P}$ to anything with the Lebesgue measure. How can I show that $\mathbb{E}(X) = \int^\infty_0 \mathbb{P}(X\geq x)$ dx with $X$ being a random variable? This must be a duplicate since the equality is very famous but I cannot find it on internet. (I only find it for discrete and continuous case) Thank you!

Answer 1

Nevermind, using the layer cake representation and Fubini's theorem, the proof follows from $$\mathbb{E}(X)=\int_\Omega \int^\infty_0 1_{X\geq x} dx d\mathbb{P}$$ Then use Fubini (X has to be non negative)