How to see Ky Fan metric satisfies the triangle inequality?
If $|X-Y| \le \epsilon_1$ and $|Y-Z| \le \epsilon_2$, then we have $|X-Z|\le \epsilon_1 + \epsilon_2$.
Hence if $|X-Z| > \epsilon_1 + \epsilon_2$, then $|X-Y| > \epsilon_1$ or $|Y-Z|> \epsilon_2$.
$$\{ \omega:|X(\omega) - Z(\omega)| > \epsilon_1 + \epsilon_2\} \subseteq \{\omega:|X(\omega)-Y(\omega)| > \epsilon_1\} \cup \{\omega:|Y(\omega)-Z(\omega)| > \epsilon_2\}$$
Hence, by the union bound,
$$P(|X-Y| > \epsilon_1) + P(|Y-Z| > \epsilon_2) \ge P(|X-Z| > \epsilon_1 + \epsilon_2)$$