Measure of correlation between two random variables

Question

I have two random variables, $X$ and $Y$ that are jointly distributed according to CDF $F(.,.)$ and a pdf $f(.,.)$ with support $[0,1]\times [0,1]$. Moreover, the marginal distributions, $F_X(x):= \int_0^1 f(x,y) dy$ and $F_Y(y)$ (defined analogously) coincide.

I want to define the following notion of correlation or interdependence between $X$ and $Y$.

$(X,Y)$ are more correlated than $(\hat X,\hat Y)$ if,

1. The marginal distributions of $X,Y,\hat X,\hat Y$ coincide.
2. And, for a.e. $x$ and interval $I$ such that $x \in I$,

\begin{align*} \mathbb P(Y \in I \vert X = x) \ge \mathbb P(\hat Y \in I \vert \hat X = x) \end{align*}

I am unable to find any order that says something like this. Does anyone know if this notion has been used by anyone? Any references would be great.