Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable.


We define the distribution function of $X$ by 


$$F(x) = P(X \leq x)$$


and the quantile function of $X$ by 


$$Q(\alpha)= \inf \{x \in \mathbb R : F(x) \geq \alpha \}$$


Is it true that we always have $Q \in L^2(]0,1[)$ ? Meaning that 


$$\int\_0^1 Q(\alpha)^2 \, d\alpha < \infty$$ 


It is true for the "famous" distributions like the normal distribution, or the Laplace distribution. And it is clear that when $X$ is not $L^1$, it does not hold (Cauchy distribution).



Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?
Let $F^{-1}:=Q$. It is [well known](https://en.wikipedia.org/wiki/Inverse_transform_sampling#Definition)  that $Y:=F^{-1}(U)=Q(U)$ equals $X$ in distribution, where $U$ is any random variable uniformly distributed on $(0,1)$. Therefore, for any real $p>0$, 
$$\int\_0^1|Q(u)|^p\,du=E|Y|^p=E|X|^p.$$
So, $Q\in L^p((0,1))$ iff $X\in L^p(\Omega,\mathcal A,P)$, for each real $p>0$. 


