it is well known, that $d(X,Y) := \mathbb{E} [ \sup_{t \in [0,T]} |X_t - Y_t | \land 1 ]$ describes a metric and satisfies $X^n \rightarrow^{ucp} X$ $\iff$ $d(X^n,X) \rightarrow 0$ („meterable“)
Does someone know a metric d, where the second equivalency / statement doesn‘t hold. As a hint: $d$ must be chosen, so that there is a sequence $X^n$ with $d(X^n,X) \rightarrow 0$, but $X^n$ doesn‘t convergent ucp.
Definition ucp: $X^n_t \rightarrow^{ucp} X_t$ if $\sup_{t \in [0,T]} |X^n_t - X| \rightarrow^{n \rightarrow \infty} 0$ in probability