Distance between two Random Variables by comparing Cumulative Distribution Functions Suppose $X$ and $Y$ are two random variables. Define the distance between $X$ and $Y$, $d(X, Y)$ as: $$d(X, Y) = \int_{-\infty}^{\infty}\left|\mathbb{P}(X < t) - \mathbb{P}(Y < t)\right|dt$$
whenever this integral makes sense. Does this distance have a name? (Or, do you know of any similar constructions?) I am interested in examples for which the total variation distance is large but this distance is not so large.
 A: It seems that, for any distributions $\mu$ and $\nu$,
$$
d(\mu,\nu)=\inf\{\mathbb E(|X-Y|)\mid \mathbb P_X=\mu,\mathbb P_Y=\nu\}.
$$
This is called the Wasserstein distance (for the $L^1$ distance) , or the Monge-Kantorovich-Rubinstein metric, or some other name.
By comparison, the total variation distance $d_{TV}$ is defined as
$$
d_{TV}(\mu,\nu)=\inf\{\mathbb P(X\ne Y)\mid \mathbb P_X=\mu,\mathbb P_Y=\nu\}.
$$
If $\mu$ and $\nu$ are measures on the integers, using the inequality $\mathbb 1_{x\ne y}\leqslant|x-y|$ for integers $(x,y)$, one sees that $d_{TV}\leqslant d$ (but that no inequality $d\leqslant c\cdot d_{TV}$ can be valid). 
For measures on the real line, no inequality $d_{TV}\leqslant c\cdot d$ can be valid, as the example of Dirac measses at $x$ and $y$ shows, when $x-y\to0$.
A: It looks very close to what is called the total variation distance  between two probability measures.
A: In the paper "Calculation of the Wasserstein Distance Between Probability Distributions on the Line", it is shown that your distance is precisely the Wasserstein metric with $p=1$, if your space is the real line.
In particular, given two probability measures $P_X$ and $P_Y$ on $\mathbb R$ (with corresponding CDFs $F_X$ and $F_Y$), the Wasserstein metric (with $p=1$) becomes:
$$
W_1(P_X,P_Y)=\int_{-\infty}^{+\infty}|F_X(t)-F_Y(t)|\,dt,
$$
which is exactly your measure. The result can be extended to probability measures defined on $\mathbb R^n$.
Moreover, if your space is bounded in $\mathbb R$, $W_1$ metrizes weak convergence. That is, letting $\rightharpoonup$ denote weak convergence, we have:
$$
P_n\rightharpoonup P\qquad\text{if and only if}\qquad \lim_{n\rightarrow\infty}W_1(P_n,P)=0.
$$
A: I do not know any special name for that function; in any case in order to satisfy the property
$$d(X,Y)=0 \Leftrightarrow X=Y$$
one needs to consider the equivalent classes of continuous random variables which are equal in distribution. On other constructions: usually divergences are used to introduce distance-like measure of distances between random variables. You can check "Methods of Information Geometry" by Amari and Nagaoka for all constructions and definitions. If you are searching for a distance (in the pure mathematical sense), then probably you should have a look at Information Value and Hellinger distance.
