Almost Sure Convergence of Random Elements of a Metric Space

In the setting of Probability Spaces: Notation, let $(S,d)$ be a metric space, let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of random elements of $S$, and let $X$ be a random element of $S$.

(i) Almost sure convergence. $(X_{n})$ converges to $X$ almost surely if there is an event $A$ with $P(A)=1$ such that, for every $\omega\in A$, the sequence $(X_{n}(\omega))_{n\in\mathbb{N}}$ converges to $X(\omega)$ in $(S,d)$.