I have a question to the following theorem (or corollary?):
Let X be a real valued r.v. and $\phi$ a measurable function from $\mathbb{R} \rightarrow \mathbb{R}$, such that $\phi \mathring{} X \in L(P)$ integrable. Let $F$ be the distribution function of the measure $P^X$ of $X$ and $F^{-1}$ be the generalized inverse. Then:
$$E[\phi(X)] = \int_{(0,1)} \phi(F^{-1}(t)) d\lambda (t)$$
If Ishow/know that the following equality holds:
$$E[X] = \int_0^1 F_X^{-1}(y)dy$$, is it possible to show the above statement somehow? Or is there another way to prove it?