Infimum of the Ky Fan metric achieved

Question

We have a probability space: $ (\Omega, \mathcal{A}, P)$ and random Variables $X,Y$. We define the Ky Fan metric as $$ d(X,Y):= \min \{\epsilon\ge 0\mid P(|X-Y|>\epsilon) \le \epsilon\} $$

My question: Why is the infimum of the Ky Fan metric achieved and so the definition well defined?

$T:=\{\epsilon\ge 0| P(|X-Y|>\epsilon) \le \epsilon\}$

I define a monoton decreasing sequence $ \epsilon_k \rightarrow d(X,Y) (k \rightarrow \infty)$ with $\epsilon_k \in T$.

$P(|X-Y|> \epsilon_k) \le \epsilon_k \le \epsilon_j$ for all $ j <k$.

For all $ j<k$ we have :

$ P(\{\omega: |X(\omega)-Y(\omega)| > \epsilon_j)\}) \le P(\{\omega:|X(\omega)-Y(\omega)| >\epsilon_k)\}\le\dots \le P(\{\omega: |X(\omega)-Y(\omega)| > d(X,Y))\})$

Answer 1

Let $F$ be the cumulative distribution function of $\left|X-Y\right|$. With the notations of the opening post, we have for each $l$: $$\tag{*} 1-F\left(\varepsilon_l\right)\leqslant \varepsilon_l.$$ Now, since the sequence $\left(\varepsilon_l\right)_{l\geqslant 1}$ is non-increasing, we get by right continuity of a cumulative distribution function that $\lim_{l\to +\infty}F\left(\varepsilon_l\right)=F\left(d\left(X,Y\right)\right)$. We conclude by taking the limit as $l$ goes to infinity in (*).