\begin{definition}[Definition:Random Variable/Discrete/Definition 1]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. 
Let $\struct {S, \Sigma'}$ be a measurable space. 
A '''discrete random variable''' on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$ is a mapping $X: \Omega \to S$ such that:
:$(1): \quad$ The image of $X$ is a countable subset of $S$
:$(2): \quad$ $\forall x \in S: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$
Alternatively, the second condition can be written as:
:$(2): \quad$ $\forall x \in S: X^{-1} \sqbrk {\set x} \in \Sigma$
where $X^{-1} \sqbrk {\set x}$ denotes the preimage of $\set x$.
\end{definition}