Measuring the distance between two probability measures using quantile functions? There are many metrics on the space of probability measures on $(\mathbb R, \mathcal B)$. 
Most of the famous metrics use the distribution functions associated with the probability measures to compute their distance. But some laws do not have an explicit form of distribution function, but they do have a nice quantile function. Are there any famous metric on the space of probability measures on $(\mathbb R, \mathcal B)$ (you can add $L^p$ regularity if needed) that are based on quantile functions instead of distribution functions ?
 A: The Wasserstein-2 metric for univariate probability measures can be formulated in terms of quantile functions. If $P_1$ and $P_2$ are probability measures with quantile functions $q_1$ and $q_2$, the squared Wasserstein distance between them is
$$ W^2(P_1,P_2) = \int_0^1 (q_1(u)-q_2(u))^2du. $$
For a proof, see Peterson and Muller, "Wasserstein covariance for multiple random densities". 
A: Updated to reflect @dave's observation:
For continuous probability distributions, defined on a connected set in $\mathbf R$ (eg. an interval $[a,b]$, $\mathbf R$ or one of the half lines $\mathbf R_{\leq 0}, \, \mathbf R_{\geq 0}$) we can show that pointwise convergence of the quantile function is equivalent to convergence in distribution.
To see this we note:


*

*For continuous distributions, the quantile function $Q$ is the inverse of the CDF $F$.

*$Q$ and $F$ are monotonic increasing functions.

*Connectedness of the domain (support) of the distribution ensures the functions are strictly monotonic - and hence bijective.
Together these are sufficient conditions for pointwise convergence of a sequence $Q_n \rightarrow Q$ to imply pointwise convergence of their inverses (see here), $F_n \rightarrow F$, which is the definition of convergence in distribution.
