Definition

random element

Let $d$ be a metric on a set $S$. Let $\mathcal{O}$ be the collection of all open sets in the metric topology on $S$ (i.e. $\mathcal{O}$ is the collection of all unions of $\epsilon$-balls in $S$). Let $\mathcal{S} = \sigma(\mathcal{O})$. If $X: \Omega \to S$ has $X \in \mathcal{B}/\mathcal{S}$ then we call $X$ a random element of $S$.