\begin{definition}[Definition:Random Vector]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $n \in \N$. 
Let $\struct {S_1, \Sigma_1}$, $\struct {S_2, \Sigma_2}$, $\ldots$, $\struct {S_n, \Sigma_n}$ be measurable spaces.
Let: 
:$\ds S = \prod_{i \mathop = 1}^n S_i$
For each integer $1 \le i \le n$, let $X_i$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S_i, \Sigma_i}$. 
Define a function $\mathbf X : \Omega \to S$ by: 
:$\map {\mathbf X} \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_n} \omega}$
for each $\omega \in \Omega$. 
We call $\mathbf X$ a '''random vector'''.
\end{definition}