I am stucked in showing an equality concerning the expectation of a fonction $f \in \mathcal{L}_1(P) , ||f||_1 = \int_R |f(x)|dP(x)$ , where $P$ is a probability measure.
The exercise is the following:
Let $Q_f $ be the inverse functions of the càdlàg function $ t \to \mathbb{P}\left( |f(X)|> t\right)$ , where $\mathcal{L}(X)=P$.
Show that $ ||f||_1 = \int_0^1 Q_f(u)du$
Does anyone has any advice?