I am trying to prove that for a random variable $X$,
$$E(X)=\int_{[0,\infty)}P(X\geq x)dx$$
I thought about first proving it for simple functions and then using monotone convergence to get nonnegative random variables, and finally, any random variable. But I am stuck already on the step of simple functions. In the case that $X$ is an indicator random variable, it is easy, but to get all simple functions, I need the RHS of the equation to be linear. I can show it respects scaling, but I can't get additivity. That is to say, I want to show that for random variables $X$ and $Y$,$$\int_0^\infty P(X+Y\geq x)dx=\int_0^\infty P(X\geq x)dx+\int_0^\infty P(Y\geq x)dx.$$If I could prove this, I think I can prove the rest.