Krdytkyu

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.

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
$$
d_{TV}(mu,nu)=inf{mathbb P(Xne Y)mid mathbb P_X=mu,mathbb P_Y=nu}.
$$
If $mu$ and $nu$ are measures on the integers, using the inequality $mathbb 1_{xne y}leqslant|x-y|$ for integers $(x,y)$, one sees that $d_{TV}leqslant d$ (but that no inequality $dleqslant ccdot d_{TV}$ can be valid).

For measures on the real line, no inequality $d_{TV}leqslant ccdot d$ can be valid, as the example of Dirac measses at $x$ and $y$ shows, when $x-yto0$.

It looks very close to what is called the total variation distance between two probability measures.

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.

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:
$$
P_nrightharpoonup Pqquadtext{if and only if}qquad lim_{nrightarrowinfty}W_1(P_n,P)=0.
$$